Surrogate optimization of variational quantum circuits

Significance

Optimization on quantum hardware is required in many applications, including Hamiltonian simulation to quantum machine learning; this entails interesting problems that must be addressed both for noisy near-term and fault-tolerant hardware. We developed a method leveraging classical computing to accelerate optimization algorithms, which can vastly reduce the required quantum resources. We use the surrogate optimization technique; we learn approximate cost function landscapes utilizing cutting-edge classical simulation. We apply our approach to state preparation for chemical and condensed matter, showing speedup in all cases. We demonstrate we can use high-performance classical computing to better understand quantum advantage for optimization acceleration on quantum hardware.

Abstract

Variational quantum eigensolvers are touted as a near-term algorithm capable of impacting many applications. However, the potential has not yet been realized, with few claims of quantum advantage and high resource estimates, especially due to the need for optimization in the presence of noise. Finding algorithms and methods to improve convergence is important to accelerate the capabilities of near-term hardware for variational quantum eigensolver or more broad applications of hybrid methods in which optimization is required. To this goal, we look to use modern approaches developed in circuit simulations and stochastic classical optimization, which can be combined to form a surrogate optimization approach to quantum circuits. Using an approximate (classical central processing unit/graphical processing unit) state vector simulator as a surrogate model, we efficiently calculate an approximate Hessian, which is passed as input for a quantum processing unit or exact circuit simulator. This method will lend itself well to parallelization across quantum processing units. We demonstrate the capabilities of such an approach with and without sampling noise and a proof-of-principle demonstration on a quantum processing unit utilizing 40 qubits.

Preparing accurate ground states for quantum systems on quantum hardware is crucial for the fields of high-energy and atomic physics (1–14), biology (15, 16), medicine (17–20), condensed matter (21–35), and quantum chemistry (21, 36–61). The large unknown costs of constructing these states on both near-term and fault-tolerant hardware is an open area of research (62–65). While fault-tolerant hardware is still in development, it is critical that short-depth efficient circuits for state preparation are available. The variational quantum eigensolver (VQE) has shown promise for moderately sized systems to efficiently construct the states where the circuits are optimized on the quantum hardware.

VQEs are not without their faults; as the number of parameters increases, these algorithms can become costly and can have exponential scaling in the worst situations, such as in cases in which barren plateaus form (66–70). Other components of the algorithm must be carefully considered, such as numerical gradient evaluations (71, 72), and choice of ansatz (73–78). Many ideas exist to improve and accelerate the reach of VQEs as it is an active field of research (36, 79–84).

An option to bypass these issues is to adopt an approach in which no optimization is performed on the quantum hardware (85). One uses heuristic classical computational approaches to optimize the parameters of a quantum circuit before running on actual hardware. Many different approaches can be realized, one of which has been extensively explored with chemistry applications using approximate circuit simulators using up to 64 qubits with moderate computational resources (86, 87).

The no-optimization approach then leaves the question of how to go beyond the classically optimized circuit effectively to find quantum advantage. In particular, after optimizing classically as much as possible, what is the best way to go forward and optimize the circuit further with quantum resources? To answer this question, we take inspiration from a geometry optimization algorithm developed for noisy electronic structure calculations (88). In ref. 88, the authors use a surrogate model (in this case, density functional theory) to calculate a Hessian for the density functional potential energy surface, which is expected to be near the exact minimum geometry. The approximate Hessian is then used to provide conjugate directions for a line search using a noisy evaluation routine (diffusion Monte Carlo calculations); each line search in a given conjugate direction uses a fixed number of points. In the following work, we translate this to circuit optimization for quantum computing applications. We describe an approach that uses an approximate circuit simulator (86, 87) to obtain an approximate Hessian for the full VQE simulation.

Variational algorithms are ubiquitously important in classical computing, which can be seen extensively in the explosion of machine learning techniques. Variational algorithms are also fundamental in many quantum techniques, which include ideas for algorithms in state preparation and quantum machine learning. Therefore, it remains an important task to utilize existing tools and techniques to develop novel engineering approaches for optimization of quantum algorithms on hardware. One approach is to push classical computation to accelerate quantum computing (36, 89–91). In this work, we show how to apply the surrogate Hessian algorithm to the optimization of quantum circuits. Case studies for a selection of molecules and quantum spin models will demonstrate this approach’s effectiveness, including using an IBM quantum computer for the transverse Ising model using 40 qubits.

Theory and Models

In this work, we employ an optimization procedure laid out in ref. 88, which optimizes an accurate and noisy cost function with the help of a cheap and noiseless surrogate. The surrogate model is an approximation, but it provides reasonable guesses for the most important features of the optimization landscape, such as a starting point and a Hessian at the minimum. Eigenvectors of the (surrogate) Hessian form a basis of conjugate directions along which the optimization can be solved instantaneously, in principle. Each of the conjugate directions can be optimized in parallel, and thus, a multi-parameter optimization of any size can be treated in a fast-converging series of parallel iterations. Typically, if the surrogate is a good approximation to the high-level function, only a few parallel iterations are necessary. The conjugate directions are solved using a line-search approach, which is robust to noise and independent of gradients: The function parameters are offset by a regular grid of $N$ shifts along the line and then evaluated using the high-level theory. The minimum is located by fitting a low-order polynomial, and the parameters are shifted accordingly. The surrogate Hessian is kept fixed throughout the iteration. The next iteration uses the same conjugate directions and search window sizes since the Hessian of the low-level theory is fixed. The entire procedure is illustrated in Fig. 1. The method stands out from the usual optimization methods (92) by being derivative-free and noise-robust, and drawing profound benefits from the surrogate model, including the Hessian and minimization of the statistical cost to meet certain accuracy requirements. Before delving deeper into the surrogate method, it is worth highlighting the difference between this method and earlier line search works. For example, the sequential minimal optimization (SMO) (93) and the related Rotosolve and its generalizations (94, 95) use a fitting procedure across the circuit parameter directions. In the SMO approach, the parameters are updated one at a time, while in this work, they are updated all at once, in parallel.

Fig. 1.

Diagram of the surrogate line search optimization procedure. The energy surfaces shown correspond to a 2D projection of the energy surface for the Ising model studied in this work. The feedback loop between the classical computer and the quantum processing unit (QPU) or quantum computer repeats for as many iterations as desired.

The low-level surrogate, which is used to compute the Hessian in this work, has multiple forms, including a sparse wave function simulator (86, 96) and a tensor network simulator (34). The choice of efficient surrogates is expected to be problem-dependent. Electronic structure calculations where wavefunctions are likely to be highly entangled but sparse will benefit from a sparse wavefunction simulator. Certain low-dimensional models may have dense wavefunctions but possess significantly lower entanglement, in which case an matrix product state (MPS) simulator would be more beneficial. The low-level Hessian computed with our sparse wave function simulator (SWS) takes advantage of the sparsity of the electronic wave function to simulate chemical systems of up to 64 qubits that would otherwise be too costly (86). The SWS makes large calculations tractable by truncating the wave function, which is controlled by two input parameters, $N_{\mathrm{MAX}}$ and $N_{\mathrm{CUT}}$. After applying each operator in the factorized unitary coupled cluster ansatz (96), we truncate the wave function to keep only the $N_{\mathrm{CUT}}$ determinants, or computational basis states, with the largest magnitudes of the wave function amplitudes if the wave function contains more than $N_{\mathrm{MAX}}$ determinants. This approximate treatment allows us to keep the significant contributions of the wave function and maintain a computationally tractable calculation. This type of approximation has been used to develop classical algorithms previously (65, 97–101). In addition, we allow for sampling noise to be included in the simulations.

For this work, we look at two classes of models: second quantized electronic Hamiltonians and the transverse field Ising model. The molecular Hamiltonians are written in the form:

$$\begin{array}{c}\hfill \widehat{H}={\displaystyle \sum _{p,q}^{\mathrm{orb}.}}{h}_p^q{\widehat{a}}_p^{†}{\widehat{a}}_q+\frac{1}{2}{\displaystyle \sum _{p,q,r,s}^{\mathrm{orb}.}}{h}_{\mathit{pq}}^{\mathit{rs}}{\widehat{a}}_p^{†}{\widehat{a}}_q^{†}{\widehat{a}}_r{\widehat{a}}_s,\end{array}$$ [1]

where the coefficients $h_p^q,$ and $h_{\mathit{pq}}^{\mathit{rs}}$ are a function of the constituent atoms and their positions. The primary molecules investigated were H₂O and N₂ in the STO-3G basis, and H₄ chain and N₂ in the cc-pVDZ basis. The ${\mathrm{H}}_4$ simulations use an intermolecular distance of 1.27$\mathrm{\AA }$ while all other molecular structures are taken from the National Institute of Standards and Technology computational chemistry database (102). The H₄ molecule uses a stretched interatomic distance because it is a more strongly correlated distance. We provide information regarding the surrogate parameters in Table 1.

Table 1.

Surrogate parameters for noisy simulations

Molecule	Basis	$N_{\mathrm{PAR}}$	$N_{\mathrm{QUB}}$	$N_{\mathrm{CUT}}$	$N_{\mathrm{MAX}}$	$N_{\mathrm{TOT}}$	$N_{\mathrm{shots}}(\delta E={10}^{-3})$	$N_{\mathrm{shots}}(\delta E={10}^{-4})$	$N_{\mathrm{shots}}(\delta E={10}^{-5})$
${\mathrm{H}}_2\mathrm{O}$	STO-3G	28	14	10	10	1,225	$7\times {10}^5$	$7\times {10}^7$	$7\times {10}^9$
${\mathrm{N}}_2$	STO-3G	55	20	60	80	14,400	$4\times {10}^6$	$4\times {10}^8$	$4\times {10}^{10}$
${\mathrm{N}}_2$	cc-pVDZ	50	36	200 to 500	200 to 500	$3\times {10}^7$	$2\times {10}^8$	$2\times {10}^{10}$	$2\times {10}^{12}$
${\mathrm{H}}_4$	cc-pVDZ	193	40	50	50	36,100	$4\times {10}^5$	$4\times {10}^7$	$4\times {10}^9$

$N_{\mathrm{shots}}(\delta E)$ corresponds to the number of shots per function call required for a desired energy precision. $N_{\mathrm{PAR}}$ is the number of parameters in the ansatz, $N_{\mathrm{QUB}}$ is the effective number of qubits, and $N_{\mathrm{TOT}}$ is the total number of wavefunction determinants.

The transverse field Ising model we investigated in this work has a Hamiltonian of the form:

$$\begin{array}{c}\hfill \widehat{H}=J_1{\displaystyle \sum _{i=1}^{N_s}}{\widehat{\sigma }}_i^z{\widehat{\sigma }}_{i+1}^z+J_2{\displaystyle \sum _{i=1}^{N_s}}{\widehat{\sigma }}_i^z{\widehat{\sigma }}_{i+2}^z+h_t{\displaystyle \sum _{i=1}^{N_z}}{\widehat{\sigma }}_i^x.\end{array}$$ [2]

For this model, we set the parameters $J_1=1.0$, $J_2=0.9$, $h_t=0.4$, and the number of sites, $N_s=40$, with periodic boundary conditions. As shown in later sections, benchmark simulations were performed for other system sizes.

Most of our simulations use the unitary coupled cluster singles and doubles ansatz,

$$\begin{array}{c}\hfill |{\mathrm{\Psi }}_{\mathrm{UCC}}⟩=e^{\widehat{T}-{\widehat{T}}^{†}}|{\mathrm{\Psi }}_0⟩,\end{array}$$ [3]

where $|{\mathrm{\Psi }}_0⟩$ is a reference state, $|{\mathrm{\Psi }}_{\mathrm{UCC}}⟩$ is the target state for the VQE optimization, and $\widehat{T}$ is the singles and doubles cluster operator:

$$\begin{array}{c}\hfill \widehat{T}={\displaystyle \sum _i^{\mathrm{occ}.}}{\displaystyle \sum _a^{\mathrm{vir}.}}{\theta }_i^a{\widehat{a}}_a^{†}{\widehat{a}}_i+{\displaystyle \sum _{i,j}^{\mathrm{occ}.}}{\displaystyle \sum _{a,b}^{\mathrm{vir}.}}{\theta }_{\mathit{ij}}^{\mathit{ab}}{\widehat{a}}_a^{†}{\widehat{a}}_b^{†}{\widehat{a}}_j{\widehat{a}}_i.\end{array}$$ [4]

The summations in the previous equation extend over the occupied and virtual orbitals. The coefficients, $\theta$, are variational parameters that are optimized so that the energy $E=⟨{\mathrm{\Psi }}_{\mathrm{UCC}}|\widehat{H}|{\mathrm{\Psi }}_{\mathrm{UCC}}⟩$ is minimized. We use the Trotterized form of Eq. 3 with the order of the operators according to the coupled cluster singles and doubles amplitudes.

Sampling Noise Studies in Electronic Structure

In this section, we present results on the feasibility and efficacy of the surrogate line search method. We use the molecular Hamiltonians from (Eq. 1) to investigate the feasibility of the surrogate line search method and its performance compared to other traditional optimizers. Furthermore, the effects of surrogate accuracy using the molecule N₂ and modifications of the line search algorithm using the H₂O molecule are examined.

In Fig. 2A, we show a comparison for noisy optimizations of N₂ in the cc-pVDZ basis using the line search and six traditional classical optimizers: SLSQP, BFGS, Powell, conjugate gradient (CG), COBYLA, and ExcitationSolve (ES) (95). All optimization alternatives start with the same initial parameters determined by the surrogate. Compared to the other choices of optimizers, the surrogate line search converges more quickly and with significantly greater accuracy. These results show that the surrogate line search offers a significant reduction in function calls compared to other classical optimizers. The ExcitationSolve algorithm does converge initially at a similar rate to the surrogate Line search for N₂ but plateaus at a higher final energy than the surrogate line search algorithm.

Fig. 2.

(A) A comparison of the performance of surrogate line search, SLSQP, BFGS, Powell, COBYLA, ExcitationSolve (ES) and conjugate gradient (CG) for the N₂ molecule in the cc-pVDZ basis using the 18 lowest orbitals and 50 terms from the CCSD expansion with the largest coefficients. The resolution on the energy is $\delta E={10}^{-5}$. (B–D) Comparison of Powell optimizer to the surrogate line search using 7 points per search direction for three molecules and bases: H₂O in STO-3G basis (Left), N₂ in STO-3G basis (Middle), H₄ chain in cc-pVDZ basis with interatomic distance 1.27$\mathrm{\AA }$ (Right). B1-D3 corresponds to truncations errors for $\delta E={10}^{-3},{10}^{-4},{10}^{-5}$ respectively. Solid markers correspond to the surrogate line search, and open markers correspond to the Powell optimizer. The details of each simulation are provided in Table 1. (E) Comparison of different values of $N_{\mathrm{CUT}}$ for a truncated ${\mathrm{N}}_2$ simulation using the parameters from Table 1 and energy uncertainties $\delta E={10}^{-5}$.

In light of these results, we investigated the efficacy of the surrogate line search compared to the noise resilient optimizer (Powell) (103–106) and ES (95), a generalization of Rotosolve and SMO (93, 94), for three molecules, H₂O and N₂ in the STO-3G basis, and H₄ in the cc-pVDZ basis. These simulations were performed using various levels of sampling noise, as discussed below. Additional simulations conducted without noise are provided in SI Appendix. Other shot-efficient optimizers do exist, such as individual Coupled Adaptive Number of Shots (iCANS) (107) as well as SHOt-Adaptive Line Search (SHOALS) (108). The iCANS and SHOALS algorithms are both shot adaptive optimizers that attempt to minimize the sampling costs depending on search directions. The benefit of this is that it will be a measurement frugal optimization process, a feature important for quantum simulations on NISQ-era hardware. The surrogate optimizer we use here, see ref. 88, can be used to adaptively weight sampling costs along various search directions similar to the iCANS and SHOALS algorithms; however, we did not implement this as the feature is still in early development and leave it as an open research question for future work.

The Powell, ES, and line search methods use initial parameters determined by the surrogate method. The initial search directions for Powell are conjugate directions determined from the approximate optimization’s Hessian. The choices for $N_{\mathrm{CUT}}$, $N_{\mathrm{MAX}}$ as well as the number of parameters $N_{\mathrm{PAR}}$, number of qubits $N_{\mathrm{QUB}}$, and total number of wavefunction determinants without truncations $N_{\mathrm{TOT}}$ are provided in Table 1. For each molecule, we consider optimization with error bars (for the energy cost function) $\delta E=\mathcal{O}({10}^{-3}),\mathcal{O}({10}^{-4}),\mathrm{and}\mathcal{O}({10}^{-5})$. The surrogate line search method is then compared with the Powell optimizer.

We show the comparison between the surrogate line search, Powell and ES optimizers in Fig. 2B–D for ${\mathrm{H}}_2\mathrm{O}$, ${\mathrm{N}}_2$, and ${\mathrm{H}}_4$. For ${\mathrm{H}}_2\mathrm{O}$, we find that the line search converges within three iterations for $\delta E={10}^{-3}$ but at least four iterations for higher sampling rates. At least 607 function calls using the surrogate line search algorithm are required to begin converging. In contrast, the Powell optimizer takes between 830 and 850 function calls. Unsurprisingly, as the sampling rate increases, both the Powell and surrogate line search accuracies improve. The Powell optimizations converge after two iterations and agree with the line search. Due to the greater number of function calls, later portions of the Powell optimizations are not shown. The ES algorithm takes a similar number of function calls to converge for H₂O as the line search, but it does not converge with as precise an energy uncertainty, leading to larger fluctuations in the energy estimations but nevertheless is moderately competitive with the line search.

The ${\mathrm{N}}_2$ molecule has almost double the number of parameters as the ${\mathrm{H}}_2\mathrm{O}$ molecule and shows a starker contrast between the two optimization methods. Each optimizer requires more function calls because the number of parameters has increased. As seen in the figures, both methods converge within 2 to 3 iterations; however, the Powell optimizer takes approximately 5,000 to 8,000 function calls to converge, whereas the line search only takes 1,500 to 2,000, which is a 2.5 to 4 times reduction in cost. The energy can increase because all the search directions are updated simultaneously, as seen in this example. Improved techniques might involve updating search directions sequentially rather than simultaneously. Testing such an approach will be studied in future work, and the implications for parallelizing the line searches must be considered. A striking result for the ES optimizer is that it struggles to converge beyond the ${10}^{-3}$ threshold for all sampling rates considered. This could be a possible bottleneck for this optimizer.

The stretched ${\mathrm{H}}_4$ simulation results are also quite stark and are shown in Fig. 2D. We expect that the stretched H₄ has difficulty with convergence, as it is likely a more strongly correlated system. Convergence to the desired result also happens within 2 to 3 iterations, requiring between 2,000 and 4,000 function calls. In contrast, the Powell optimizer takes nearly 6,000 to 9,000 function calls; this is a cost savings of approximately three times by using the surrogate line search approach. Compared to line search, the ES optimizer appears to converge faster and with larger uncertainties. However, it again struggles to meet higher-precision targets, $\delta E={10}^{-4},{10}^{-5}$, and instead the convergence plateaus.

Until now, tunable parameters for the surrogate line search have been fixed for specific problems. As problem sizes are scaled up, it is important to consider how the accuracy of the surrogate will impact the efficacy of the surrogate line search. It will be expected that if the surrogate is too coarse, then changes to the potential energy surface will be quite discontinuous, and estimations of the Hessian, and by analogy, the conjugate search directions, will be negatively affected; in principle, the numerically calculated Hessian could have negative eigenvalues falsely indicating that the system is not in a local minimum. We show the effects of a line search using values of $N_{\mathrm{CUT}}=200,225,250,300,400,\text{and}500$ using the cc-pVDZ basis for the N₂ in Fig. 2E. To keep resource costs manageable for this molecule, we use only the 18 lowest orbitals and keep only 50 operators for the UCC ansatz corresponding to the terms from the coupled cluster singles and doubles expansion with the largest coefficients. For suitably large values of $N_{\mathrm{CUT}}>300$, we find that the Hessian is accurate enough to provide a suitable surrogate line search and generally converges to a within $2\times {10}^{-3}$ Hartree. However, there is a break-over point, $N_{\mathrm{CUT}}<300$, where the Hessian is ill-defined and does not provide accurate search directions. The inaccurate search directions are a by-product of the Hessian being discontinuous in parameter space for the surrogate model. This is a place where surrogate optimization can break down in terms of feasibility, especially at larger scales. However, one could possibly mitigate the issues with a non-positive definite Hessian by various corrections and methods that have been considered in the literature for Newton and quasi-Newton methods, such as damped least square, where a scalar times the identity is added to the Hessian to enforce greater numerical stability. These approaches can be tested further for surrogate approaches as we push towards systems that are close to our limits of classical simulation.

Another question we can ask is whether we can reduce the required function calls by limiting the line searches to a subset of the steepest search directions. This type of modification to the surrogate optimization algorithm is an entirely new investigation. We look specifically at the STO-3G basis ${\mathrm{H}}_{2}O$ simulation for this case. The Hessian calculated using SWS has 10 eigenvalues around 80 and 19 eigenvalues around 5; a larger eigenvalue indicates a steeper search direction. We limit the surrogate line search to the conjugate directions corresponding to the 10 largest eigenvalues. A comparison of this truncated search with the entire search in Fig. 3 using a fixed energy error $\delta E={10}^{-4}$. The truncated search method converges with fewer function calls than the full line search. Because the truncated search direction optimization has a lower energy than the full search at early times, this suggests that one could dynamically include search directions as the optimization improves.

Fig. 3.

Comparison between the surrogate line search using the 10 steepest of the 29 total set of search directions for H₂O in the STO-3G basis versus all available search directions with energy precisions $\delta E={10}^{-4}$. Points displayed are the infinite sampling limit and error bars are calculated using statistical bootstrap from the uncertainties on the parameters. The blue line indicates the optimal result for the truncated search directions.

Transverse Ising Model Using a Quantum Computer

We used the transverse Ising model from (Eq. 2) as a prototypical example for simulations on quantum processing units (QPUs). We used this Hamiltonian as an example case to avoid issues measuring long correlated strings of Pauli matrices, such as those appearing in the fermion-to-qubit mapping for quantum chemistry problems. We use a nearest-neighbor ansatz, which is easily implemented on many quantum platforms. This ansatz uses repeated layers of entangling rotations given by the operator

$$\begin{array}{c}\hfill {\widehat{U}}_{i,j}(\theta )=e^{-i\theta ({\widehat{\sigma }}_i^y{\widehat{\sigma }}_j^z+{\widehat{\sigma }}_i^z{\widehat{\sigma }}_j^y)}.\end{array}$$ [5]

The explicit form of the ansatz used in our work is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \widehat{V}(\overrightarrow{\theta })& =(\prod _{i=0}^{N_s/2}{\widehat{U}}_{2i,2i+1}({\theta }_1))(\prod _{i=0}^{N_s/2}{\widehat{U}}_{2i+1,2i+2}({\theta }_2))\hfill \\ \hfill & \times (\prod _{i=0}^{N_s/2}{U}_{2i,2i+1}({\theta }_3))(\prod _{i=0}^{N_s/2}{\widehat{U}}_{2i+1,2i+2}({\theta }_4)).\hfill \end{array}\end{array}$$ [6]

The ansatz acts upon the computational state: $|0⟩{\otimes }^{N_s}$. We specifically chose this operator as its structure is similar to a UCC-style ansatz and forces the coefficients of the wavefunction to be real, an expected result for the ground state. We performed a surrogate line search on IBM’s ibm_brisbane. The system we simulate uses the parameters $J_1=1.0$, $J_2=0.9$, $h_t=0.4$, and $N_s=40$ and is in the gapped phase but is close to a phase transition.

The surrogate model was a MPS simulator whose maximum bond dimension was set to 4. After calculating the Hessian using the MPS simulator, we found that two search directions are relatively shallow with eigenvalues of $3\times {10}^{-2}$ and $4\times {10}^{-2}$. For this reason, we chose to perform the line search across the two steepest search directions. The results from the QPU after three iterations are shown in Fig. 4. Although the energies extracted from the simulation are well above the expected values, the parameters have converged to 2.5 SDs of the expected result and, in the absence of noise, give results within 1% of the ansatz minimum.

Fig. 4.

40 qubit simulation of the transverse field Ising model on ibm_brisbane using the ansatz in Eq. 6 using only 2 steepest search directions. The expected QPU points are the results of an MPS simulation with bond dimension 400 using the parameters determined from the corresponding QPU simulation.

Several error mitigation strategies have been leveraged for these simulations, including dynamical decoupling (DD) (109–114), randomized compilation (RC) (115–122), and Clifford rescaling (123–125). A brief overview of these methods can be found in SI Appendix. These simulations show that decoherence and amplitude damping are substantial noise sources and that Clifford renormalization does not account for all errors. Each simulation used 20k shots for each basis measurement in the Hamiltonian.

To have confidence in the validity of any simulations, we compared the energy calculated using a variational circuit on a QPU to an MPS representation of the corresponding state for lattices with 12, 20, 24, 28, and 32 sites. We calculate the energy of the ibm_brisbane QPU using the parameters corresponding to the surrogate minimum. The discrepancy between the surrogate result and various error mitigation techniques is shown in Fig. 5, including probabilistic error cancelation (PEC) (126, 127) and zero noise extrapolation (ZNE) (126, 128, 129), which were not included in the production calculations. These results are discussed further in SI Appendix. There are linear trends in the divergence of the QPU simulations from the expected results. Since the circuit depth does not increase with increased system size, this indicates that the primary source of systematic error is either coherent or depolarizing errors from the entangling gates as opposed to decoherence effects from the qubit lifetime.

Fig. 5.

Energy of the transverse Ising model calculated on ibm_brisbane using dynamic decoupling (DD), readout error mitigation (readout), zero noise extrapolation (ZNE), and probabilistic error cancellation (PEC), compared to MPS simulator with bond dimension 40 with parameters fixed from the potential minimum of the surrogate model.

Outlook

This work shows that the surrogate Hessian line search method from (88) is amenable to the optimization of variational quantum eigensolver circuits in chemistry and condensed matter. The line search method has been shown to work both for hardware efficient and unitary coupled cluster type ansätze. In particular, this new optimizer outperforms the Powell optimizer by a factor of 2 to 4 in the cases we studied in the presence of sampling noise. The efficacy of this line search optimization is contingent upon accurate conjugate search directions being obtained from Hessian in the low-level theory; in particular, if negative eigenvalues appear in the surrogate’s Hessian due to discontinuities in parameter space, the line search algorithm struggles to perform an adequate optimization. We also investigated the effects of truncating the search directions using only the steepest conjugate directions. We found that the line search can optimize the circuit faster than using all the line search directions at the expense of some degree of accuracy. In addition, a proof of principle demonstration was performed on ibm_brisbane for a transverse Ising model using 40 qubits.

While the studies here investigate standard example cases, this method opens up the possibility of investigating frustrated quantum spin systems, which are challenging for both tensor network simulators and quantum Monte Carlo methods. Example models encompass a broad range, for example, including Kitaev spin liquids (27, 28, 130–132), geometrically frustrated Kagome antiferromagnets (133–135), and downfolded models of molecules (136).

It is important to investigate methods to mitigate the problems that arise from ill-defined Hessian directions. Is it possible to truncate search directions whose Hessian eigenvalues are negative or near zero as a first-pass optimization and then use a traditional optimizer or a modified version of the line search to improve the optimization? Additionally, the optimization procedure may be robust to quantum gate noise; while some preliminary studies are provided in SI Appendix, more detailed studies across a wide range of models and ansätze will be essential to confirm this. If true, this would indicate that surrogate line searches will be a promising method toward quantum advantage in variational quantum algorithms. Other avenues could include incorporating this optimization with other derivative-free methods (137) as well as combining techniques of gradient optimization with extra function calls to the QPU. This work highlights the possibility of quantum advantage with variational methods. Our approach may be compatible with many different classical simulation techniques that can be used as a surrogate. We have shown here that by leveraging state-of-the-art classical methods with high-performance computing resources, we can provide a starting point for QPU optimization results that can be accelerated to quantum advantage. In fact, our work circumvents many bottlenecks for quantum optimization, including sampling resources. This comes from the fact that the sampling resources scale quadratically with respect to the inverse of the estimated uncertainty, but by fitting to multiple points along the potential energy surface, we can reduce the target uncertainty compared to traditional fixed-point energy estimations.

However, the benefits of our work extend beyond the scope of quantum optimization algorithms; one could feasibly look at the original scope of the surrogate optimization algorithm from ref. 88 and look at replacing the choice of diffusion Monte Carlo with estimations on a quantum computer. This lends itself well to fault-tolerant quantum simulation, where performance guarantees could be made. In total, this shows that the benefits of our work extend beyond the field of variational quantum optimization and into electronic structure calculations as a whole.

Materials and Methods

The underlying optimizer we use for surrogate line search is available at the following https://github.com/QMCPACK/stalk/releases/tag/v0.1 originally developed by authors J. Krogel and J. Tiihonen. The optimizations methods work out of the box as is and the only components that are necessary for inclusion are the surrogate and stochastic potential energy cost functions. The surrogate software and certain instances of the full stochastic wavefunction were implemented using the sparse wavefunction simulator (86) which uses trunctation methods to remove low weight determinants in the wavefunction to keep the total memory requirements for simulation of chemical systems within a reasonable size this involves setting an upper limit $N_{\mathrm{MAX}}$ which triggers a procedure to remove the total number of determinants in the wave function down to $N_{\mathrm{CUT}}$. Meanwhile the simulations for the transverse Ising model used the IBM QISKit development environment with the matrix product state simulator as the surrogate simulator and ibm_brisbane quantum computer as the actual cost function. The simulations on the IBM quantum computer used a combination of SE mitigation techniques as well as a rescaling technique which uses Clifford circuits which can be classically simulated to estimate what the noiseless value should have been for a certain class of circuits where all parameters are 0, which we denote $⟨O_{Exp.}⟩$. Given a depolarizing error model it is expected that the observed observable, $⟨O_{Obs.}⟩$, should have the following relation:

$$\begin{array}{c}\hfill ⟨O_{Obs.}⟩=\gamma ⟨O_{Exp.}⟩.\end{array}$$ [7]

We measure $⟨O_{Exp.}⟩$ classically and $⟨O_{Obs.}⟩$ on the quantum computer to experimentally determine the value $\gamma$. This value is then used to “rescale” any observations on future QPU runs to counteract the depolarizing noise on the quantum computer.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Data for figures in the main paper is provided in SI Appendix and at ref. 138.