Introduction

Quantum computing has opened up a realm of possibilities, especially in the area of simulating and understanding complex quantum systems in areas like quantum chemistry and condensed matter physics. Recent paper by QunaSys describes a promising approach- Quantum-Selected Configuration Interaction (QSCI) algorithm. It offers a novel and more efficient way to find ground-state and excited-state energies in quantum systems. In particular, the new approach circumvents effects of the errors that can spoil the variational nature of VQE: that is, the energy estimated by quantum devices is not guaranteed to give an upper bound on the exact ground-state energy. This is problematic because lowering the resulting energy of VQE does not necessarily mean approaching to the exact ground state. As we will learn QSCI solves this problem and yields results obeying the variational principle. In this blog post, we’ll delve into the key components and advantages of the QSCI algorithm, and how it is paving the way towards useful quantum computing. Below we describe the algorithm in general terms.

Algorithm Overview

1. Start with an Approximate Ground State: The QSCI journey begins with the preparation of an approximate ground state. Quantum computers have the unique capability to generate these states, often using techniques such as Variational Quantum Eigensolvers (VQE) or other
   quantum algorithms. This is the foundation on which QSCI builds. Note that our recent algorithm, ADAPT-QSCI, enables us to avoid the dependence on VQE. This will be discussed in the next article.
2. Identifying Crucial Configurations: Once we have our approximate ground state, the next step is to identify the vital computational basis states or electron configurations that are essential for expressing the ground state accurately. This is where the quantum magic happens. The quantum state is repeatedly measured to pinpoint these key configurations, a task that would be incredibly challenging for classical computers.
3. Truncated Hamiltonian Matrix Diagonalization: With the important configurations in hand, it’s time to perform the diagonalization, but here’s the twist — it happens on classical computers. The Hamiltonian matrix, which describes the energy of the quantum system, is truncated to include only the identified configurations. This considerably reduces the computational workload, making it classically tractable. As the final result one obtains the smallest eigenvalue and its corresponding eigenvector, which approximates the ground-state energy.
4. Extending the Algorithm for Excited States: QSCI isn’t limited to ground state calculations. It can also be used to find excited states by expanding the subspace or by repeating the procedure for each energy eigenstate. This versatility is a significant advantage of the algorithm.
5. A Strict Upper Bound on Ground-State Energy: The Hamiltonian matrix elements in the computational basis can be accurately calculated using classical computers. Therefore, the diagonalization step yields an energy value that serves as a strict upper bound on the exact ground-state energy. Regardless of the quality of the subspace spanned by the identified configurations, this bound remains valid. The quality mainly affects how close the bound is to the exact value.
6. Mitigating Errors with Symmetries: In quantum systems with symmetries and conserved quantities like particle number, the post-selection of computational basis states in the sampling outcome can help mitigate bit-flip errors. This adds robustness to the algorithm in the
   presence of physical and statistical errors.
7. Beyond Ground State - Quantum Eigenstate Tomography: QSCI isn’t just about ground state calculations. It can be advantageous for eigenstate tomography, allowing classical estimation of the expectation values of various observables at no additional quantum cost. This efficiency stems from having a classical representation of the state, which enables the computation of expectation values efficiently.

Conclusion

In the world of quantum computing, the Quantum-Selected Configuration Interaction (QSCI) algorithm stands out as a stepping stone approach for finding ground-state and excited-state energies efficiently. Its ability to harness quantum computing for identifying crucial configurations and then performing classical diagonalization reduces computational complexity, making it an attractive tool for tackling complex quantum systems. Moreover, its utility extends beyond ground state calculations, offering a versatile solution for eigenstate tomography. As quantum computing continues to evolve, QSCI is poised to play a pivotal role in solving challenging quantum problems.

Researchers at QunaSys Inc., Osaka University, and Mitsubishi Chemical Co. have developed an algorithm for calculating the gradient of energy of molecules with a method called state-averaged orbital-optimized variational quantum eigensolver (SA-OO-VQE), which gives a well-balanced description of multiple electronic states and expected to be utilized in a variety of quantum chemistry applications of quantum computers. They have applied the energy derivative calculated based on the developed method to the problem of determining the reaction pathway of cis-trans photoisomerization including a conical intersection, whose analysis requires a balanced description for the ground and excited states, with classical numerical simulation of quantum computers. The results were published in the Journal of Chemical Theory and Computation, a peer-reviewed journal of the American Chemical Society.

“Analytical Energy Gradient for State-Averaged Orbital-Optimized Variational Quantum Eigensolvers and Its Application to a Photochemical Reaction”,

Keita Omiya*, Yuya O. Nakagawa*, Sho Koh, Wataru Mizukami, Qi Gao, and Takao Kobayashi,

Available at: https://doi.org/10.1021/acs.jctc.1c00877

(Preprint version: https://arxiv.org/abs/2107.12705)

Background

In recent years, the development and research of quantum computers have been vigorously pursued. In particular, a type of quantum computer that is now being realized, called a Noisy Intermediate-Scale Quantum (NISQ) device, is expected to outperform existing classical computers in certain tasks [1]. Because NISQ devices are subject to noise, they can only run a few relatively simple algorithms, of which one called the variational quantum eigensolver (VQE) has received the most attention [3]. VQE is a method for calculating the energies of molecules and matter based on quantum mechanics, and it is wisely designed to divide-and-distribute tasks for quantum and classical computers so that it can be easily run on NISQ devices.

In order to make the calculation of molecular energies using VQE more practical, a method called orbital-optimized VQE (OO-VQE) was proposed by several groups including QunaSys [2]. OO-VQE was further extended by combining it with a technique called state-average, and a method called state-averaged OO-VQE (SA-OO-VQE) was proposed [4]. SA-OO-VQE is expected to be applied to various chemical calculations as a method to obtain the ground state (most stable state) and excited state energies of molecules in a balanced manner using a small quantum computer.

Problem

Energy is definitely a central quantity that determines the static properties of matter, but it is also very important to know the dynamic properties, for example, how chemical reactions occur. One of the most fundamental quantities for studying dynamic properties is the derivative of energy, that is, how much the energy changes when the parameter of the system is slightly changed. For example, if we know how much the energy changes when the nucleus of a molecule moves slightly, we can find out how the molecule moves afterward.

Although the SA-OO-VQE described above can calculate the energy value itself, there is no known method to obtain its analytical derivative. For NISQ devices, which inevitably contain noise and fluctuation in outputs, calculating the derivative using finite numerical differences is considered to be inaccurate. Therefore, it is vitally important to develop an algorithm to calculate the analytical derivative of the energy.

Method and Results

In this study, we derived a formula for calculating the analytical derivative of the energy in SA-OO-VQE based on the Lagrangian method, which has been used in quantum chemical calculations using classical computers. We showed that the analytical derivative of the energy obtained by SA-OO-VQE can be calculated by appropriately combining quantities that can be easily measured on the NISQ device.

To demonstrate one example usage of this method, we have calculated the SA-OO-VQE energies and their derivatives for a model molecule that exhibits cis-trans photoisomerization including a conical intersection, which is a typical and important photochemical reaction (Fig. 1). The obtained energies and their derivatives were used as a subroutine in an existing method of finding chemical reaction pathways, and it was confirmed that the reaction paths agree well with those obtained by using the SA- CASSCF method, which is the counterpart of SA-OO-VQE on classical computers. This result exemplifies that the calculation of energy and its derivative using SA-OO-VQE can be applied to the analysis of actual photochemical reactions.

The numerical calculations in this study were performed by simulating an ideal quantum computer using a classical computer, and the energies and their derivatives using SA-OO-VQE were calculated by Qamuy, commercial software for quantum chemical calculation by quantum computers developed by QunaSys.

Outlook

This research has enabled us to calculate a fundamental quantity, the derivative of energy, in a sophisticated method using a quantum computer (SA-OO-VQE). Since the energy derivative is a quantity that is frequently used in computational chemistry, such as in the analysis of chemical reactions, the algorithm developed in this study is expected to further expand the applications of NISQ devices.

Fig. 1: A model molecule for cis-trans photoisomerization.

Fig. 2: Potential energy surfaces around the conical intersection between two electronic states involved in the photochemical reaction. Left (right) panel is a result obtained by using SA-OO-VQE (SA-CASSCF).

Fig. 3: Plot of energies along the reaction path. Left (right) panel is a result obtained by using SA-OO-VQE (SA-CASSCF).

References

[1] F. Arute et al., Nature 574, 505–510 (2019), https://www.nature.com/articles/s41586-019-1666-5

[2] A. Peruzzo et al., Nat. Commun. 5, 4213 (2014) https://www.nature.com/articles/ncomms5213

[3] T. Takeshita et al., Phys. Rev. X 10, 011004 (2020)

https://journals.aps.org/prx/abstract/10.1103/PhysRevX.10.011004,

I. O. Sokolov et al., J. Chem. Phys. 152, 124107 (2020), https://aip.scitation.org/doi/10.1063/1.5141835,

W. Mizukami et al., Phys. Rev. Research 2, 033421 (2020),

https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.2.033421,

[4] S. Yalouz et al., Quantum Sci. Technol. 6 024004 (2021), https://iopscience.iop.org/article/10.1088/2058-9565/abd334

Estimation of expectation values is a key ingredient in variational quantum algorithms [1], regarded as promising candidates for industrial applications of near-term quantum computers. A challenge in the standard ways of expectation value estimation is the necessity to repeat measurement many times to suppress statistical fluctuation, especially in the application to quantum chemistry, which typically demands high accuracy. We proposed [2] an alternative method to estimate the expectation values, which can be more efficient for a certain type of quantum state, frequently encountered in systems of quantum chemistry and condensed matter physics.

The development of Noisy Intermediate-Scale Quantum (NISQ) devices has been rapidly progressing in recent years. Promising applications of NISQ devices are variational quantum algorithms, in particular Variational Quantum Eigensolver (VQE) [3]. VQE can be used to obtain approximations to the lowest energy and the corresponding quantum state (ground state) for electrons in an atom or a molecule and can be applied to quantum chemistry, condensed matter physics, etc.

VQE is a hybrid quantum-classical algorithm based on the variational method: for a given trial wavefunction and Hamiltonian (the quantum-mechanical operator for energy) of some physical system of interest, quantum computers are used to estimate the energy expectation value; the latter is minimized with respect to parameters describing the trial wavefunction by use of a classical computer; this enables us to iteratively find the approximate lowest-energy state within a set of states described by the trial wavefunction. In this algorithm, the task for quantum computers is to estimate the expectation value. To do so, the trial wavefunction is prepared as a quantum circuit, for which “energy” is measured, and such a set of preparation-measurement procedures is repeatedly performed for statistical estimation. Since the number of measurements is finite, the estimated expectation value is inevitably accompanied by statistical fluctuation. It is highly desirable to suppress the amount of statistical fluctuation with as few measurements as possible in order to perform VQE calculations with high precision and efficiency.

The standard way of expectation value estimation [3] goes like this: firstly, the Hamiltonian is decomposed into quantities that can be measured by NISQ devices (specifically, the quantities are so-called Pauli strings, tensor products of Pauli operators); then, each of these quantities is estimated by measuring a quantum circuit; finally, recombining the estimates of those quantities, the overall estimate of the energy expectation value is obtained. The procedure is rather simple but relies on potentially huge statistical resources: estimating each of Pauli strings requires many measurements to suppress statistical fluctuation; besides, the number of Pauli strings to be measured increases significantly with the system size [4]; then, the total number of measurements can be too huge to finish the expectation value estimation within a practically acceptable time, especially when high accuracy is required as in quantum chemistry. There are vigorous efforts to improve this deficit, e.g., by simultaneously measuring commuting Pauli strings and/or by optimizing the allocation of measurement budget to each of Pauli strings. Yet, further improvements are still awaited toward practical use for systems of industrial interest. (See [5] for a recent assessment of measurement resources in the application to quantum chemistry calculations.)

We pursued another direction in expectation value estimation: instead of decomposing the Hamiltonian into measurable quantities, our method relies on the decomposition of the wave function into computational basis states. Specifically, we express the expectation value of the Hamiltonian H for a wave function |ψ> as

where the computational basis states, for N qubits, are labeled by m, n = 0, 1, …, 2ᴺ -1 (for details, see Eq. (6) and explanation around it in [2]). Our method is based on the insight that a wave function of interest in quantum chemistry and condensed matter physics can often be well-described by a relatively small number of electron configurations, each of which can be identified as a computational basis state on quantum computers. We call such a state a “concentrated” state in particular computational basis states. With this insight, we approximated the expectation value above as the sum of contributions from those major electron configurations or computational basis states. The major basis states can be selected by sampling the wave function, generated on quantum computers, in the computational basis. The transition matrix elements <m|H|n> can be efficiently calculated by a classical computer, while the remaining pieces in the above equation can be estimated based on quantities measured on quantum computers, each of which requires a quantum circuit consisting of at most N CNOT gates in addition to the ones for the wave function |ψ>. The point is that the number of quantities to be measured is O(R), where R means the number of the major basis states. Hence, when R is significantly smaller than the number of Pauli strings in the decomposition of the Hamiltonian, our method is expected to be more efficient than the standard methods in terms of statistical convergence.

In order to assess the performance of our method, we took small molecules such as water and methane as an example and considered the expectation value estimation for their electronic Hamiltonian and ground states. Calculating the variance of the energy expectation value for each molecule, we evaluated the number of measurements to suppress the amount of statistical fluctuation below the required accuracy (1 mHartree), which is then compared with the ones evaluated for the standard methods. The result is summarized in the following figures, which include the standard methods with basic implementation (No grouping) and with improvements by grouping mutually commuting Pauli strings (QWC and GC), for two ways of the measurement allocation to each of the measured quantities.

The result tells that (1) the numbers of measurements can be generically reduced by several orders of magnitude in comparison with “No grouping”, (2) our method is generically better than “QWC” (grouping of qubit-wise commuting Pauli strings), (3) our method tends to be comparable with “GC” (grouping of general-commuting Pauli strings). We remark that the GC grouping requires more expensive quantum circuits [6] than our case, i.e., O(N²) (GC) vs. O(N) (ours) extra two-qubit gates in addition to the circuit for the wave function, and hence is more challenging to implement on noisy devices.

In summary, we proposed an alternative way for expectation value estimation, and demonstrated that our method can be more efficient than the standard methods in terms of statistical convergence. Our method can be incorporated in VQE as a subroutine, which may speed up VQE. The method can be applied to expectation value estimation of generic observables, and thus can be used in combination with variational quantum algorithms other than VQE. It is open to a wide range of applications.

References and footnote

[1] For review, see, e.g., Suguru Endo, Zhenyu Cai, Simon C. Benjamin, Xiao Yuan, “Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation”, J. Phys. Soc. Jpn. 90, 032001 (2021). [https://journals.jps.jp/doi/10.7566/JPSJ.90.032001]

[2] M. Kohda, R. Imai, K. Kanno, K. Mitarai, W. Mizukami, and Y. O. Nakagawa, “Quantum expectation value estimation by computational basis sampling”, arXiv:2112.07416. [https://arxiv.org/abs/2112.07416]

[3] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O’Brien, “A variational eigenvalue solver on a photonic quantum processor”, Nat. Commun. 5, 4213 (2014). [https://www.nature.com/articles/ncomms5213]

[4] For instance, the number of the Pauli strings scales as O(N⁴) for a Hamiltonian of electronic state in a molecule, where N is the number of spin-orbitals or qubits. It is only polynomial in the system size, but can be numerically large, given a nominal target of quantum computations would be N≈100 or beyond.

[5] Jérôme F. Gonthier, Maxwell D. Radin, Corneliu Buda, Eric J. Doskocil, Clena M. Abuan, Jhonathan Romero, “Identifying challenges towards practical quantum advantage through resource estimation: the measurement roadblock in the variational quantum eigensolver”, arXiv:2012.04001. [https://arxiv.org/abs/2012.04001]

[6] Pranav Gokhale, Olivia Angiuli, Yongshan Ding, Kaiwen Gui, Teague Tomesh, Martin Suchara, Margaret Martonosi, Frederic T. Chong, “Minimizing State Preparations in Variational Quantum Eigensolver by Partitioning into Commuting Families”, arXiv:1907.13623. [https://arxiv.org/abs/1907.13623]

Written by Masaya Kohda.

Introduction

For the past few years, we have witnessed the advent of near-term quantum computers, or noisy intermediate-scale quantum (NISQ) devices [1]. NISQ devices consist of several tens or hundreds of qubits that are not equipped with error correction and are prone to noise. Owing to these reasons, NISQ devices are not suitable for performing complicated quantum algorithms such as Shor’s algorithm for prime factorization. However, they still allow us to tackle problems that cannot be handled by classical computers and have been extensively studied by both researchers in academia and those in commercial companies.

One of the promising “killer applications” of NISQ devices can be found in the field of quantum chemistry and condensed matter physics, where researchers try to reveal the nature of molecules and solid materials based on quantum mechanics. While calculations by classical computers become exponentially time-consuming as the size of a target system increases, quantum computers can handle the system only in a polynomial amount of computational resources. This unique characteristic of quantum computers (and NISQ devices) attracts a lot of attention from the field of material research for various industrial purposes such as the development of batteries and organic electro-luminescence molecules.

Although a lot of methods have been developed aiming to leverage NISQ devices in investigating quantum systems, there has still been one missing building block: the Green’s function (GF). GF is a cornerstone to understanding quantum systems and tells us a variety of properties of them. For example, GF informs us of what kinds of excitations are present in the target system, which is the clue to predict how the system responds to an external field such as electric field and heat. Despite its importance, however, the NISQ-friendly methods for calculating GF had barely been developed before Ref. [2].

Methods to calculate Green’s function on near-term quantum computers

In Ref. [2], methods for calculating Green’s function on near-term quantum computers were proposed by Suguru Endo, Iori Kurata (former interns at QunaSys Inc.), and Yuya O. Nakagawa (Lead researcher at QunaSys Inc.). In that work, two different approaches were proposed. The first approach makes use of the variational quantum simulation (VQS) algorithms [3] to calculate GF directly, while the second approach adopts the so-called Lehmann representation of GF with the subspace-search variational quantum eigensolver (SSVQE) [4] or the multistate-contracted variational quantum eigensolver (MCVQE) [5].

Let us introduce a definition of GF (in the simplest form)

, where H is the Hamiltonian of the system,

is the annihilation (creation) operator of electrons at site i, and |G〉 is the ground state of the system.

For a given initial state |Φ〉 and an ansatz quantum circuit U(θ) (realized on NISQ devices), the original VQS algorithm finds optimal parameters θ* such that U(θ*)|Φ〉becomes the closest to the time-evolved state

Naive application of the original VQS to the calculation of GF requires the circuit depicted in Fig. 1. This circuit contains two controlled-U gates that should be very complicated and are not expected to run on NISQ devices.

The authors of Ref. [2] extended VQS such that U(θ*)|Φ> and U(θ*)P|Φ> simultaneously approximate

and

, where P is some Pauli-gate. The quantum circuit to evaluate GF by using the extended VQS technique is shown in Fig. 2. A variational circuit U(θ(t)) which simultaneously approximates the time evolution operator for the initial states |G⟩ and P_j |G⟩ is constructed, instead of two different controlled-unitary operations. This circuit may be feasible on NISQ devices.

On the other hand, when we calculate GF by the second approach based on SSVQE or MCVQE, we utilize the spectral function representation of GF, or the Fourier transform of GF, defined as

where E_G is the energy of the ground state and E_n is the n-th excited state.

SSVQE and MCVQE yield energies of the ground state and excited states as well as their transition amplitudes

, so simply plugging in those values obtained by SSVQE and MCVQE into the equation above is enough to calculate the spectral function representation of GF.

Feasibility estimation for two-dim. Hubbard model of 25 site

The authors also provided the resource estimation to calculate GF of the two-dimensional Hubbard model of 25 sites, which is almost impossible to solve exactly by classical computers. According to the estimation, the acceptable fidelity of two-qubit gate operations is 0.1% to calculate GF with the method based on VQS (the first approach) under some assumptions. This value of the fidelity is within the reach of the state-of-the-art NISQ devices [6].

Summary and outlook

In summary, the methods for calculating GF on near-term quantum computers were proposed by researchers at QunaSys Inc. and collaborators. Since GF is vital to exploring quantum systems, various applications of the methods are expected. For example, they can be used as the solver for the hybrid approach of dynamical mean-field theory [7] or slave boson methods [8] for strongly correlated systems, which are notoriously difficult to be tackled with classical computers. The methods to calculate GF on NISQ devices will broaden our horizon of quantum computing and consequently engineering applications, widening our understandings of more complicated electron systems.

Written by Reika Fujimura and Yuya O. Nakagawa.

Reference

[1] J. Preskill, “Quantum Computing in the NISQ era and beyond”, Quantum 2, 79 (2018).

[2] S. Endo, I. Kurata, and Y. O. Nakagawa, “Calculation of the Green’s function on near-term quantum computers” , Phys. Rev. Research 2, 033281 (2020).

[3] Y. Li and S. C. Benjamin, “Efficient Variational Quantum Simulator Incorporating Active Error Minimization”, Phys. Rev. X 7, 021050 (2017).

[4] K. M. Nakanishi, K. Mitarai, and K. Fujii, “Subspace-search variational quantum eigensolver for excited states”, Phys. Rev. Research 1, 033062 (2019). See also this blog post.

[5] R. Parrish et al., “Quantum Computation of Electronic Transitions Using a Variational Quantum Eigensolver”, Phys. Rev. Lett. 122, 230401 (2019).

[6] F. Arute et al., “Quantum supremacy using a programmable superconducting processor”, Nature 574, 505–510 (2019).

[7] B. Bauer et al., “Hybrid Quantum-Classical Approach to Correlated Materials”, Phys. Rev. X 6, 031045 (2016).

[8] Y. Yao et al., “Gutzwiller Hybrid Quantum-Classical Computing Approach for Correlated Materials”, Phys. Rev. Research 3, 013184 (2021).

QunaSys keeps developing efficient quantum algorithms to accelerate various applications of quantum computers. Our mission is to enthusiastically develop technologies that bring out the maximum potential of quantum computers and to continually deliver innovations to society.

Introduction: the advent of quantum computers and machine learning

For some problems in physical science, we need to take care of quantum information to compute or simulate target physics because it is described by quantum mechanics. In most cases, however, it is hard to handle such quantum information on classical computers. Given that, it is natural to think of processing quantum information as it is by a quantum computer, like Feynman’s famous quote, “Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical”[1]. Today, we are witnessing the realization of prototypical digital quantum computers; quantum devices consisting of hundreds to thousands of non-fault tolerant qubits (quantum bits), the so-called noisy intermediate-scale quantum (NISQ) devices, have been developed rapidly in the past few years. Some commercial companies start providing access to their early-stage NISQ devices to their customers, although the devices seem still too small and noisy to execute any industrially-competitive tasks.

Meanwhile, the rapid growth of machine learning algorithms has been applied to various fields in physical science, allowing us to solve problems that have no analytical solution and require a huge amount of computational resources even with numerical methods. So, even though the allowed computations on the NISQ devices are quite limited, combining machine learning algorithms with the quantum devices may circumvent such computational burdens for solving quantum problems, and they may accelerate the applications of the NISQ devices to reach the industrial level.

Background: solving quantum chemistry by a quantum machine

Quantum chemistry, which studies properties of chemical materials based on quantum mechanics, is one of those fields with wide applications of machine learning. The main task of quantum chemistry is to solve the Schrödinger equation, a fundamental equation of quantum mechanics, to predict various chemical phenomena such as chemical reactions. Although many computational methods for solving the equation on classical computers have been developed for decades, they often require gigantic computational resources, which prevent us from investigating many interesting and industrially-important processes such as photo-excitation dynamics of molecules.

One of the promising ways to make use of the NISQ devices for quantum chemistry is the so-called variational quantum eigensolver (VQE), which is a variational algorithm finding the ground state (the most stable state) of a given quantum system [2]. The key idea of the VQE is to combine a programmable and parametrized quantum circuit implemented on the NISQ devices with a classical optimization technique. By optimizing the quantum circuit classically, the VQE algorithm can find the ground state of the system with only a shallow quantum circuit that is presumably executable even on the NISQ devices.

The ground state does not describe the whole physics of the quantum system, but the excited states are also essential to analyze it. Nonetheless, even though extensions of the VQE to find excited states have been proposed recently by several researchers [3], they usually require a larger number of runs of the quantum circuit than the VQE does, and therefore it is not so easy to run them on the real NISQ devices.

Our proposal: machine learning of "quantum" data

To present another approach to find the excited states of quantum systems, Hiroki Kawai (a former intern at QunaSys Inc.) and Yuya O. Nakagawa (a lead researcher at QunaSys Inc.) proposed a simple quantum machine learning scheme, or a machine learning algorithm utilizing a quantum computer [4]. Their proposal aims to learn physical quantities from quantum data. Its capability was demonstrated to predict the properties of electronic excited states of small molecules. The model is trained with supervised machine learning from only its ground state wavefunction realized as a qubit state on the NISQ devices. It has the following distinct features:

・It learns wavefunctions of the target system that are "quantum" data. Most quantum machine learning algorithms treat classical data such as text, images, etc., and have the difficulty of encoding classical data in quantum states on quantum computers. Learning from quantum data is a natural setup of performing quantum machine learning, and there is no overhead in encoding the data into quantum states because the data are already quantum.
・Quantum information should give a more accurate description of the molecular system than classical information, which is so far used for classical machine learning methods applied to quantum chemistry.
・ It reduces the cost of finding the excited states because it only requires the quantum data of the ground state obtained by the usual VQE whose resource requirements are less than finding the excited states by the extensions of the VQE.
・It may be executable on the NISQ devices since the required circuit depth is shallow.

Let us describe the details of the algorithm. As well as the VQE, it has both the quantum and classical parts (Fig. 1). First, we input a quantum state that is the ground state of the target Hamiltonian, which is computed by the VQE beforehand. The quantum part consists of a random quantum circuit called quantum reservoir (specified as U_ent in Fig. 1) and the single-qubit Pauli measurements. The reservoir, which can be any kind of random quantum circuit and is fixed during the learning, increases the entanglement of the input quantum state so that one can obtain non-local information from only the measurements of the local, single-qubit Pauli operators (we note that quantum reservoir was introduced in [5]). The measurements in the quantum part of the algorithm give a 3N-dimensional classical real-valued vector

where N is the number of qubits. Conducting only the single-qubit Pauli measurements makes it easy to implement the algorithm on the near-term NISQ devices. Moreover, as few as three different circuits are needed to obtain the vector x since the single-qubit operators acting on different qubits commute with each other, and hence the measurements of them can be performed simultaneously. After obtaining the classical vector, the algorithm proceeds to the classical part which is a simple classical machine learning unit, e.g. linear regression. The classical machine learning unit is trained to predict the excited-state properties of the target Hamiltonian from this vector x. More details can be found in the original paper [4].

Numerical Results

In [4], numerical simulations for small molecules (LiH and H_4) with various geometric structures were performed to demonstrate the applicability of the method to problems in quantum chemistry.

The Hamiltonians for electronic states of the two molecules are considered. The first and second excited energies of the Hamiltonian and the transition dipole moment from the ground state to the first excited state are chosen as the target excited-state properties that the proposed algorithm seeks to predict from the ground state wavefunctions. The authors numerically simulated two situations; one is the noiseless situation, where the ideal outputs from the quantum circuits are available, and the other is the noisy situation, where the quantum circuit has noise and the measurement results have statistical and systematic errors.

The prediction results for the LiH molecule as an example are shown in Fig. 2. We can see that the proposed algorithm predicts the exact values of the excited-state properties with high precision for the noiseless cases. Even in noisy cases resembling the actual NISQ devices, it reproduces the approximative values.

Conclusion

In conclusion, Kawai and Nakagawa introduced a new quantum machine learning scheme for predicting the excited-state properties of a molecular Hamiltonian from its ground state wavefunction. It consists of a random quantum circuit called a quantum reservoir and a simple classical machine learning unit. The numerical simulations demonstrated that it can accurately predict the target excited-state properties and has the potential to be implemented on a near-term NISQ device. Although numerical demonstrations were performed only for the small molecules, we expect that it is applicable to larger molecules appearing in more practical applications in quantum chemistry and material science in the future.

References

[1] Feynman R P, International Journal of Theoretical Physics 21 467–488 (1982)
[2] Peruzzo A, McClean J, Shadbolt P, Yung M H, Zhou X Q, Love P J, Aspuru-Guzik A and O’Brien J L, Nature Communications 5 4213 (2014)
[3]
McClean J R, Kimchi-Schwartz M E, Carter J and de Jong W A, Phys. Rev. A 95, 042308 (2017),
Colless J I, Ramasesh V V, Dahlen D, Blok M S, Kimchi-Schwartz M E, McClean J R, Carter J, de Jong W A and Siddiqi I, Phys. Rev. X 8, 011021 (2018),
Nakanishi K M, Mitarai K and Fujii K, Phys. Rev. Research 1, 033062 (2019), see also this blog post,
Parrish R M, Hohenstein E G, McMahon P L and Martınez T J, Phys. Rev. Lett. 122, 230401 (2019),
Higgott O, Wang D and Brierley S, Quantum 3 156 (2019),
Jones T, Endo S, McArdle S, Yuan X and Benjamin S C, Phys. Rev. A 99 062304 (2019),
Ollitrault P J, Kandala A, Chen C F, Barkoutsos P K, Mezzacapo A, Pistoia M, Sheldon S,Woerner S, Gambetta J and Tavernelli I, Phys. Rev. Research 2, 043140 (2020),
Tilly J, Jones G, Chen H, Wossnig L and Grant E, Phys. Rev. A 102, 062425 (2020)
[4] Kawai H and Nakagawa Y O, Mach. Learn.: Sci. Technol. 1, 045027 (2020)[5] Fujii K and Nakajima K, Phys. Rev. Applied 8, 024030 (2017)

Written by Hiroki Kawai and Yuya O. Nakagawa.

QunaSys keeps developing efficient quantum algorithms to accelerate various applications of quantum computers. Our mission is to enthusiastically develop technologies that bring out the maximum potential of quantum computers and to continually deliver innovations to society.

While classical computers have bits representing either 0 or 1 values, quantum computers have qubits which provide arbitrary superposition of the bases |0〉and |1〉of quantum states. This intrinsic difference forms the powerfulness of quantum computers overwhelming classical computers. Though the quantum computer’s full potential is still in progress, we are now in the first stage where the so-called NISQ (Noisy Intermediate-Scale Quantum) devices are available. NISQ devices can operate only the limited number of algorithms for quantum computers, or quantum algorithms, because the error in their qubits prevents us from executing the complicated quantum algorithms. In the coming 3–5 years, we will need efficient quantum algorithms to explore the quantum advantage with NISQ devices having 50 to a few hundred qubits.

One of the most practical applications of quantum computers is expected to happen in quantum chemistry. This is because simulating molecules based on quantum mechanics requires exponential computational resources with increasing the number of atoms in molecules on classical computers, whereas it does polynomial on quantum computers. In particular, an algorithm called variational quantum eigensolver (VQE) [1] was designed to find energies of a target molecule and has shown its potential to run on real NISQ devices.

The VQE enables us to calculate electronic energies of molecules. When analyzing molecules in quantum chemistry, however, the derivatives of the energies with respect to some parameters defining the system are also essential. For example, the derivative with respect to the nuclear coordinates of the molecules gives the force acted on them. The force is utilized in various ways, such as determining the most stable molecular structure, elucidating chemical reaction paths, and performing molecular dynamics simulations. Another example is the derivative with respect to the external electric or magnetic fields, which tells us the spectroscopic properties of the molecules when they are irradiated by light.

There are two ways to obtain the energy derivatives in quantum chemistry. One is to use the finite-difference method, where we compute the derivatives at two proximate points and calculate the difference numerically. The other is the analytical approach, where we employ some analytical formulas for the derivative. The former is always applicable without any prior knowledge of the method to calculate the energies but vulnerable to the noise in the calculated energies. Since NISQ devices contain inevitable noises in their outputs, the analytical method is more practical in the current quantum age.

In 2019, two groups independently proposed methods to calculate analytical energy derivatives based on the VQE and applicable to NISQ device [2,3]. One was from Kosuke Mitarai (assistant professor at Osaka University & CSO at QunaSys Inc.), Yuya O. Nakagawa (lead researcher at QunaSys Inc.), and Wataru Mizukami (associate professor at Osaka University) in Japan [2]. They showed analytical formulas to calculate the derivatives of the energy obtained by the VQE, up to the third-order derivatives. They explicitly provided quantum circuits and ways of measurements to determine the quantities appearing in those formulas. All procedures in the proposed method require shallow (not-complicated) quantum circuits and are expected to be feasible on real NISQ devices. It is worthwhile to note that the proposed method can be applied to the derivative of excited-state energy, which is more difficult to calculate with classical computers.

On the other hand, Tom O’Brien at Leiden University (currently also at Google Inc.) and his colleagues proposed a different method to evaluate the analytical energy derivatives on quantum computers [3]. The formula for the first-order derivative is the same as the method in Mitarai’s work, but the formulas for the second- and higher-order derivatives are based on the so-called sum-over-state approach that requires transition amplitudes involving all excited states in the system. In practice, the summation over all the excited states is truncated at the finite number of the states, and the error is introduced in the formals in a rigorous sense. Compared with Mitarai’s proposal, O’Brien’s work does not suffer from the classical computational cost related to the number of circuit-parameters used for the VQE, with a trade-off to the necessity to calculate the excited states even for evaluating the derivatives of the ground-state energy. In addition to the theoretical proposal, the authors of [3] did an experiment to calculate the first-order derivate of energy with respect to the nuclear displacements on a two-qubit superconducting device and illustrated they could perform the geometry optimization of hydrogen molecule based on that derivative.

The energy derivatives are universally used in the calculation of the properties of molecules. The proposals explained in this blog post open up the further possibility to take advantage of NISQ devices as we approach the quantum-computer era. With the simultaneous development of the hardware of quantum computers and new quantum algorithms to exploit them, we are looking forward to more and more practical applications of quantum computers to calculations in quantum chemistry, which would eventually outperform classical computers.

References

[1] A. Peruzzo, J. McClean, P. Shadbolt, M.-H Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien. “A variational eigenvalue solver on a photonic quantum processor.” Nat. Commun. 5, 4213 (2014). [link]

[2] K. Mitarai, Y. O. Nakagawa, and W. Mizukami. “Theory of analytical energy derivatives for the variational quantum eigensolver.” Phys. Rev. Research 2, 013129 (2020). [link]

[3] T. E. O’Brien, B. Senjean, R. Sagastizabal, X. Bonet-Monroig, A. Dutkiewicz, F. Buda, L. DiCarlo, and L. Visscher. “Calculating energy derivatives for quantum chemistry on a quantum computer.” npj Quantum Inf. 5, 113 (2019). [link]

Written by Youyuan Zhang and Yuya O. Nakagawa.

QunaSys keeps developing efficient quantum algorithms to accelerate various applications of quantum computers. Our mission is to enthusiastically develop technologies that bring out the maximum potential of quantum computers and to continually deliver innovations to society.

“Symmetry” appears everywhere in modern physics, and it plays a significant role in understanding various properties of a quantum system. When a quantum system has symmetry, the energy spectrum/states of the system can be classified according to the symmetry; it is known that the symmetry introduces a conserved quantity C (an observable) in a quantum system, and an eigenvalue c of C stipulates a category of symmetry. It is demanding to predict what energies/states appear in each category of symmetry. In quantum chemistry, for example, it is known that light absorption and luminescence only occur among states residing in a certain symmetric category. Thus, to understand photochemical properties of molecules, it is essential to analyze the energies/states of the molecules based on their symmetry.

Meanwhile, a near-term quantum computer called a noisy intermediate-scale quantum (NISQ) device has been recently developed. A NISQ device has been getting attraction as the first step to the realization of more powerful quantum computers. In particular, a NISQ device is expected to simulate a large-sized quantum system that is hard to compute with classical computers. The variational quantum eigensolver (VQE) is a promising algorithm for a NISQ device to compute the ground state and its energy. The VQE utilizes a parametrized quantum circuit U(θ), an ansatz circuit, to generate a trial state |ψ(θ)〉 := U(θ)|0〉, an ansatz state. We minimize the expectation value of the Hamiltonian for the ansatz state〈ψ(θ)|H|ψ(θ)〉as a cost function with respect to the parameters θ to obtain the ground-state energy.

With the growing attention toward a NISQ device, it is suggested to compute energies/states of a given quantum system in the desired category of symmetry by utilizing the modified version of the VQE [1,2], which we call the constrained VQE referring to Ref. [2]. In the constrained VQE, we penalize the deviation of the ansatz state from the desired category to obtain such energies/states with the desired symmetry. More specifically, the following two types of cost functions have been proposed:

where C is a conserved quantity, c is the desired eigenvalue of C, and the coefficient μ, which is called the penalty coefficient, represents the magnitude of the penalty in the cost function. By minimizing those cost functions with respect to θ, we may obtain the optimal parameter θ* that yields a state |ψ(θ*)> belonging to the desired category of symmetry. For example, if we want to compute the energy of the so-called S1 state (the first excited state of singlet) of a hydrogen molecule H2, we consider the electron number operator N, the total spin squared operator S², and the z-component of spin operator S_z as conserved quantities C and designate the desired category as

However, several serious problems prevent practical applications of the constrained VQE. First, the penalty coefficient μ is heuristically chosen without solid theoretical analysis. If the coefficient μ is too small, we may have wrong results; on the other hand, if μ is too large, the optimization may not be completed properly. Currently, there is no guarantee that we can obtain energies/states with the desired symmetry as a result of the VQE. Second, the performance of the two types of penalties has not been compared, leading to the inconsistent use of the two cost functions without any evidence.

In 2020, Kohdai Kuroiwa (an intern at QunaSys Inc.) and Yuya O. Nakagawa (a lead researcher at QunaSys Inc.) provided an extensive analysis of the constrained VQE to solve these problems [3]. They theoretically investigated the two types of penalties in the constrained VQE and disclosed crucial differences between them.

First, for the function

they constructed a simple formula to determine the penalty coefficient with which we can certainly obtain the desired energies/states as a result of the optimization. Conceptually, the formula is constructed by considering the condition that the sum of the expectation energy and the penalty term becomes the smallest for the desired state. Letting

be the ground-state energy and the target-state energy of the Hamiltonian H, respectively, and

be the smallest gap among distinct eigenvalues of C, we have the following formula for the penalty coefficient μ:

On the other hand, remarkably, they revealed that

does not properly work for any finite value of the penalty coefficients.

Here, we show the graphical understanding of the reason why

does not perform well in Figure 1. In the figure, we hypothetically consider an orthogonal coordinate plane consisting of the expectation values of H and the expectation values of C. A state is mapped to a point on the plane by considering the expectation values of H and C for the state. The black points represent the energy eigenstates of H, and the coordinates of the black points express the eigenvalues of H and C for the eigenstates. The cyan convex polygon corresponds to all the possible points achievable by an ansatz state. The colored curves (yellow, green, and pink) are the contours of

with small, medium-sized, and large penalty coefficients μ. In the procedure of the constrained VQE, the point where the curve touches the polygon is regarded as the optimized quantum state. If the desired point (the desired state) is on the boundary of the polygon (the bluepoint; desired point 1), the optimized point (the tangent point) approaches the desired point but never achieves it in a rigorous sense. On the other hand, if the desired point is in the polygon (the red point; desired point 2), it cannot be the tangent point, and the optimization yields a completely wrong result.

Furthermore, numerical simulations validated these theoretical analyses. The authors of [3] considered a hydrogen chain H4 where four hydrogen atoms are aligned in line with the identical bond distance 2.0 Å. The STO-3G minimal basis set was adopted to construct an 8-qubit Hamiltonian for electronic states. They computed the energy of T1 state (the ground state of triplet) by setting S² = 2 and S_z = -1 as conserved quantities. They simulated the constrained VQE with the two cost functions having a range of penalty coefficients, μ = 0.01, 0.1, 1, 10, 100. The results are shown in Table 1.

In the table, the “residuals” represents the difference between the energy expectation value for the resulting optimized state and that for the desired energy (-1.881 876 Ha) in Hartree (Ha). The result of

with

indicates that the formula leads to the correct value. Note that for large μ (μ = 10,100), the results of the optimization of

deviate from the correct value. Thus, the formula in [3] provides an appropriate penalty coefficient to ensure that we obtain the desired value. On the other hand, the accuracy of the result of

is poor for small μ and improves as μ becomes large, which is consistent with the aforementioned graphical explanation. More details and other simulations are shown in the original paper [3].

In conclusion, the analyses of penalties in the constrained VQE revealed the fundamental difference between the performances of the two cost functions

The inconsistent and heuristic use of the two cost functions will be terminated due to the theoretical superiority of

and the formula to determine the appropriate penalty coefficient. Thanks to the generality of their theoretical analysis, the results have a significant impact on many fields of quantum physics and encourage further investigation for exploiting NISQ devices with the constrained VQE.

References

[1] J. R. McClean et al., “The theory of variational hybrid quantum-classical algorithms”, New Journal of Physics, vol. 18, (2), pp. 023023, 2016.

[2] I. G. Ryabinkin, S. N. Genin and A. F. Izmaylov, “Constrained Variational Quantum Eigensolver: Quantum Computer Search Engine in the Fock Space”, Journal of Chemical Theory and Computation, vol. 15, (1), pp. 249–255, 2019.

[3] K. Kuroiwa and Y. O. Nakagawa, “Penalty methods for variational quantum eigensolver”, Phys. Rev. Research 3, 013197 (2021)

Written by Kohdai Kuroiwa and Yuya O. Nakagawa.

QunaSys keeps developing efficient quantum algorithms to accelerate various applications of quantum computers. Our mission is to enthusiastically develop technologies that bring out the maximum potential of quantum computers and to continually deliver innovations to society.

In June 2020, Honeywell became the first to commercially offer a quantum computer achieving a Quantum Volume of 64, with the launch of their System Model HØ. Following Honeywell’s announcement, we at QunaSys started thinking about what we could do with the world’s most powerful quantum computer?

In cooperation with Honeywell, we designed an experiment to calculate the photochemical response of an organic molecule, namely azobenzene, using the 6-qubit Honeywell quantum computer System Model HØ, as shown in Figure 1. Our results show a much higher level of accuracy than a typical superconducting quantum computer. The details of the results are as follows.

Oscillator strength of azobenzene

The most interesting property of azobenzene is the isomerization between trans- and cis- isomers, which can be switched by light. Based on this photo switch mechanism, azobenzene compounds are expected to have various applications, such as photopatterning, data storage, microfluid control, and drug release.

Generally when a molecule is irradiated by light, its electrons transition from the ground state to a higher energy state as a consequence, with some probability. The oscillator strength is the key physical property in determining this probability. We can use this property to calculate the population of excited states, which can then be used to evaluate the likelihood of the isomerization. Calculating the oscillator strength requires a quantum computer to have high accuracy since it needs to prepare superpositions of multiple energy-eigenstates, even if we use a proprietary hardware-friendly method (see the last section for details). Therefore, calculating the oscillator strength is an ideal test for Honeywell’s 6-qubit device that is experimentally simultaneously challenging and of practical importance. In the experiment, we calculated the oscillator strength of azobenzene using the hardware and compared the results with the exact classical simulation.

Experiment

We calculated the oscillator strength between two energy eigenstates of cis-azobenzene, using the subspace-search variational quantum eigensolver (SSVQE), proposed by QunaSys Inc. For a detailed description of the overall experimental setup, please see the last section.

Figure 3 shows the calculation result of the oscillator strength of azobenzene. As evident in the figure, the value from Honeywell’s hardware (red dot) is much closer to the exact value (horizontal dashed line) we are aiming for than the simulation result based on the noise level of a typical superconducting quantum computer (blue dot). These results indicated that the fidelities of gate operations are of essential importance when calculating more complex physical quantities.

Conclusion

We used Honeywell’s 6-qubit trapped-ion quantum computer with Quantum Volume 64 to calculate the oscillator strength of the azobenzene molecule. The accuracy of the results obtained on Honeywell’s quantum computer is much higher than those achieved through a simulation on a typical superconducting quantum computer. By combining Honeywell’s ultra-high-precision hardware and the SSVQE algorithm, we successfully calculate this important quantity with high accuracy, expanding the horizon of quantum computers’ applications. Although some have questioned the ability of trapped-ion systems to scale, the QCCD architecture realized by Honeywell has the potential to solve these bottlenecks. Honeywell has already announced a new release of a next generation system with a Quantum Volume of 128. We’re looking forward to the future of this technology which we believe will accelerate industrial applications of trapped-ion quantum computers.

Details of simulation setup

We use the subspace-search variational quantum eigensolver (SSVQE), which we proposed in [Phys. Rev. Research, 1, 033062 (2019)], in this project. The SSVQE is an extension of the variational quantum eigensolver (VQE) and finds low-lying excited states of a given Hamiltonian by optimizing a parametrized unitary U(θ) so that

where

is an arbitrary (but easily preparable) set of mutually orthogonal states and

is the j-th excited state of the Hamiltonian. As the optimization step needs many iterations to converge, we perform the optimization on a classical computer in this project. This may be avoided by using more sophisticated optimization techniques such as stochastic gradient descent which require less samples to optimize but due to time limitations, we circumvent possible trial-and-error in this step.

We use an electronic Hamiltonian of cis-azobenzene molecule, which becomes a 6-qubit Hamiltonian under an appropriate active space approximation. With this Hamiltonian, we evaluate the oscillator strength, which is proportional to a sum of the absolute square of transition amplitude

where

is a dipole moment operator between the ground state and the first singlet excited state (note: this value vanishes for trans-azobenzene due to molecular symmetry). Using the optimized unitary, we evaluate

by the following “virtual interference” technique. First, we prepare

and

Second, we measure

Finally, we obtain

by classically computing the following:

Further, to mitigate hardware errors, we employed a linear extrapolation technique.

Written by
Yohei Ibe and Tennin Yan at QunaSys Inc.
(We deeply acknowledge helpful feedbacks from Honeywell's team)

QunaSys keeps developing efficient quantum algorithms to accelerate various applications of quantum computers. Our mission is to enthusiastically develop technologies that bring out the maximum potential of quantum computers and to continually deliver innovations to society.

Quantum chemistry aims to describe the behavior of atoms and molecules. Geometrical structures, molecular vibrational motions, chemical reaction dynamics, and photon absorption and emission are critical characteristics. A complete and accurate electronic structure is a prerequisite for obtaining that information accurately. This requirement brings up the Schrodinger equation, which governs the electrons,

H|ψ⟩=E|ψ⟩.

The Hamiltonian H is unique for the target molecule. Solving the Schrodinger equation is equivalent to solving the eigenvalue problem for H, i.e., finding the eigenvectors and the corresponding eigenvalues. However, this problem becomes notoriously hard for classical computers as the size of the molecule increases. It is virtually impossible to precisely solve the Schrodinger equation because the dimension of the Hamiltonian grows exponentially as the number of electrons increases. A complete calculation of a molecule with dozens of atoms will require more classical bits (memories) than the number of atoms in the entire observed universe. However, for quantum computers, the polynomial number of “quantum bits”, or qubits, is required with the increase of the target’s size.

Just like the classical computer takes decades to show its full potential, the quantum computer is coming, but step by step. The first phase is the next three to five years, during which the so-called NISQ (Noisy Intermediate-Scale Quantum) devices are exploited. With various error mitigation methods, NISQ devices are expected to show the quantum advantage at medium size ranging from 50 to a few hundred qubits. Quantum algorithms to solve classically difficult problems with NISQ devices are demanded.

A quantum algorithm for chemistry, called variational quantum eigensolver (VQE)[1], has been developed to solve molecules’ ground states. Usually, molecules stay at their lowest energy states, so the ground state is of particular interest. The VQE utilizes a parameterized quantum circuit U(θ), also called an ansatz circuit, to generate an ansatz state |ψ(θ)⟩= U(θ)|0⟩ where |0⟩ is some reference state. The expectation value of the target Hamiltonian ⟨H(θ)⟩ = ⟨ψ(θ)| H |ψ(θ)⟩ is minimized by iterative optimization of the parameters θ to find the electronic ground state.

Nevertheless, in addition to the electronic ground state, the electronically excited states become essential for studying chemistry. Many attractive phenomena in molecules and materials involve multiple electronic states, such as light absorption and luminescence, nonadiabatic transition, and spectroscopies. This is why a method for computing excited states by NISQ devices is highly desired so that various chemical phenomena can be explored by a quantum computer.

Based on the VQE algorithm for the ground state, there is a naive way to calculation excited states using a quantum computer. Because of the orthogonality between the eigenstates of the Hamiltonian H, to find the k-th excited state is to minimize the expectation value of H under the orthogonal condition, which guarantees that the ansatz |ψ(θ)⟩ and the ground to (k-1)-th state are orthogonal to each other. In this way, the quantum circuit called “swap test” is necessary for measuring the inner product between the states and checking the orthogonality. However, the swap test brings two significant problems: the required number of qubits becomes twice as large as that of the original system and a lot of SWAP gates are needed, resulting in a deep quantum circuit. In such a case, the overall computational time becomes more prolonged, and the accumulated error will be more significant. So NISQ devices are not considered to be compatible with this method. To obtain excited states using NISQ devices, we need to develop a method without the swap test.

Extending the VQE framework to excited states without the swap test has been achieved recently. In 2018, an intern at QunaSys Inc. Ken Nakanishi, CSO at QunaSys Inc. Kosuke Mitarai, and a professor at Kyoto University (currently Osaka University) Keisuke Fujii proposed an algorithm, the subspace-search variational quantum eigensolver (SSVQE). The general idea is to search a low energy subspace by supplying orthogonal input states to the parameterized quantum circuit U(θ); since U(θ) is unitary, the output states become automatically orthogonal and we do not have to check the orthogonality by the costly swap test. The desired k-th excited state can be obtained as the highest energy state in the low energy subspace. The proposed algorithm consists of only the optimization for expectation values of the output states and does not require any ancillary qubits, making this proposal a truly near-term quantum algorithm. Here, we explain one of the variants of the SSVQE, called weighted SSVQE, which gives all the excited states up to the k-th with only one optimization procedure.

The algorithm of the weighted SSVQE for finding up to the k-th excited state can be described in two steps (see also the figure above):

1. Construct a parameterized quantum circuit U(θ) and choose input states {|ψ_j ⟩}_{j=0,1,...,k} =0 which are orthogonal with each other (⟨ψ_i|ψ_j⟩ = δ_ij).

2. Minimize the cost function L(θ) = Σ_j w_j ⟨ψ_j|U†(θ) H U(θ)|ψ_j⟩ with respect to θ, where the weight vector w is chosen such that w_i > w_j when i < j.

As we explained, the SSVQE successfully avoids the swap test by using the orthogonal input states. This strategy further benefits us in the evaluation of the transition amplitude between the obtained eigenstates; the transition amplitude |⟨ψ_i|U†(θ)AU(θ)|ψ_j⟩|^2 between two eigenstates for an observable A can be easily evaluated by applying U(θ) to superpositions of the input states like 1/√2 (|ψ_i⟩±|ψ_j⟩) and measuring expectation values of A. This advantage enables us to investigate the light-matter interaction phenomena involved in quantum chemistry.

Finally, as a simple demonstration, we implement the SSVQE to search for the ground state and the first (ionic) excited state of a hydrogen molecule. We use the minimal STO-3G basis set to construct a 4-qubit Hamiltonian H to describe the molecule. |0000⟩ and |0001⟩ are the orthogonal input states fed into the weighted SSVQE algorithm. As you can see in the figure below, the optimization of the SSVQE cost function correctly finds the exact energies of the ground and excited states. The details of the simulation can be found in

[English]
http://docs.qulacs.org/en/latest/apply/6.3_ssvqe.html
[Japanese]
https://dojo.qulacs.org/ja/latest/notebooks/6.3_subspace_search_VQE.html

More examples in details on how to apply SSVQE can be found in the original paper [2].

In conclusion, an efficient algorithm for finding the excited states of a given Hamiltonian is proposed. Unlike the existing algorithms, this method assures the states’ orthogonality at the inputs of the parameterized quantum circuit. Each of the orthogonal input states can be mapped onto one of the energy eigenstates by minimizing a carefully designed cost function by tuning the quantum circuit’s parameters. The whole algorithm is free from the costly swap test and contains only the optimization of the expectation values of several quantum states. Moreover, the transition amplitudes between the obtained eigenstates can be easily evaluated. This new method extends the VQE to excited states without sacrificing any feasibility and will become an important step to take advantage of NISQ devices in practical calculations for quantum chemistry.

Written by Youyuan Zhang, Sho Koh, and Yuya O. Nakagawa

Reference:
[1] A. Peruzzo, J. McClean, P. Shadbolt, M.-H Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).
[2] Ken M. Nakanishi, Kosuke Mitarai, and Keisuke Fujii. Subspace-search variational quantum eigensolver for excited states. Phys. Rev. Research 1, 033062 (2019).

QunaSys keeps developing efficient quantum algorithms to accelerate various applications of quantum computers. Our mission is to enthusiastically develop technologies that bring out the maximum potential of quantum computers and to continually deliver innovations to society.

“Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly, it’s a wonderful problem because it doesn’t look so easy.” A famous description said by Richard Feynman in 1982 brings us a concept of building a quantum machine to handle the non-classical computations. As Feynman points out, it is not easy. Even now, the quantum computers are limited in accuracy because they do not have error-correcting functionality and the experimental error accumulates over time without a systematic correction. The number of qubits, or quantum bits, ready for executing a non-trivial quantum algorithm is limited to a few dozen for now. We call a machine consisting of several ten or hundred qubits without error correction the noisy intermediate-scale quantum (NISQ) devices.

To make the best use of the tremendous advantage of quantum computers with a limited scale, or the NISQ devices, a quantum-classical hybrid algorithm has been proposed. In the quantum-classical hybrid algorithm, quantum computers prepare a quantum state by executing a quantum circuit instructed by classical computers and output classical data like measurement outcomes of the prepared state. Classical computers, in turn, process the output data and give feedback to quantum computers.

One of the most commonly used quantum-classical hybrid algorithms is Variational Quantum Eigensolver (VQE, learn this in https://grove-docs.readthedocs.io/en/latest/vqe.html and http://docs.qulacs.org/en/latest/apply/6.2_vqe.html), a solution for finding the eigenvalues of Hamiltonian H of a target system based on the variational principle. A typical structure includes a quantum subroutine that runs inside a classical optimization loop, a parameterized quantum circuit, and its optimization based on an observed cost function. The quantum subroutine has two fundamental steps:

1. generate an ansatz state |ψ(θ)⟩ with the parameter θ from a parameterized quantum circuit U(θ) on quantum devices;

2. measure the expectation value of the given Hermitian operator, ⟨H(θ)⟩ = ⟨ψ(θ)|H|ψ(θ)⟩.

On classical computers, the parameters θ will be optimized to minimize the ⟨H(θ)⟩ through an optimizer for non-linear problems until it converges.

Improving the efficiency of the classical optimization process is, therefore, the key to accelerating the overall performance of the VQE. Typically, the optimization is based on the gradient method (left panel of Figure 2). In the gradient method, the optimization proceeds step by step following the direction of the gradient. This method requires a lot of evaluation of the cost function, ⟨H(θ)⟩, and highly depends on the initial input; the worse the initial guess is, the slower the optimization will be. It becomes natural to wonder if we can find another method with a smaller number of evaluations of ⟨H(θ)⟩ and less initial value dependence.

In 2019, Ken Nakanishi (intern at QunaSys Inc.), Keisuke Fujii (currently Professor at Osaka University), and Synge Todo (Professor at the University of Tokyo) proposed a new optimization method for the VQE to speed up the convergence. The proposed optimization method is hyperparameter-free, has faster convergence, has less dependence on the initial choice of the parameters, and is robust against the error in the function evaluation. In this method, the optimization is performed exactly with respect to certain chosen parameters at each step by using the characteristic structure of the typical parameterized quantum circuits. In fact, as shown in the right side of Figure 2, the cost function behaves as a simple trigonometric function, or a sine curve with period 2π with respect to the parameter θ while the other parameters are fixed, when the parameterized quantum circuit U(θ) consists of a set of unitary gates exp(iθ/2*A) being subject to A*A = I. Because of this special property, we can determine the parameter θ that provides the exact minimum of the cost function by evaluating the cost function at only three independent points with respect to the parameter. By repeatedly performing this procedure, we can optimize all the parameters so that the cost function becomes as small as possible.

This optimization method, which we call NFT optimizer after the names of the authors, allows us to update each parameter with fewer measurements than the gradient-based methods. The update of the parameters is deterministic if the order of the parameters is provided, so NFT optimizer is hyperparameter-free.

As a simple demonstration of the advantage of NFT optimizer, we applied it to calculate the energy of a hydrogen molecule and compared it with other optimizers. The convergence efficiencies are compared among NFT optimizer, the Nelder-Mead optimizer, and COBYLA optimizer under the “RYRZ” variational circuit (see top left panel of Figure 3). In two bottom panels of Figure 3, the horizontal axis is the total number of evaluations of the function ⟨H(θ)⟩ during the optimization and the vertical one is the function value. The bottom left panel of Figure 3 contains the results in an ideal situation where no noise is considered, and it shows clearly that the result obtained using NFT optimizer shown as the red curve is more than two times faster than the Nelder-Mead optimizer and COBYLA optimizer.

As mentioned before, the NISQ devices suffer from noise. Is NFT optimizer better even in this case? The bottom right panel of Figure 3 contains the results of simulating a noisy situation where the value of the function fluctuates as a result of 32 shots measurements and the noise model mimicking the actual NISQ devices made of the superconducting qubits. NFT optimizer is still much more efficient and reliable than the other two optimizers.

In summary, an intern at QunaSys and collaborators have proposed a hyperparameter-free optimizer for quantum-classical hybrid algorithms using parameterized quantum circuits. NFT optimizer seems to have better convergence efficiency even in the presence of the inevitable noise in the NISQ devices. With this new method, NISQ devices will be more practical in quantum computation.

Written by Youyuan Zhang and Yuya O. Nakagawa

Reference:
[1] Ken M. Nakanishi, Keisuke Fuji, and Synge Todo, https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.2.043158
[2] Implementation by K. Nakanishi is given in https://github.com/ken-nakanishi/nftopt

QunaSys keeps developing efficient quantum algorithms to accelerate various applications of quantum computers. Our mission is to enthusiastically develop technologies that bring out the maximum potential of quantum computers and to continually deliver innovations to society.