Phase estimation using random Hamiltonian

Abstract

Existing methods for dynamical simulation of physical systems use either deterministic or random selection of terms in the Hamiltonian. In the present application, example methods are disclosed in which the Hamiltonian terms are randomized and the accuracy of the random draw approximation is adapted as the accuracy required in the phase estimate increases. This reduces the number of quantum gates required and, in some cases, the number of qubits used in the simulation.

Description

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0002] This application claims priority to U.S. Non-Provisional Application No. 16 / 430,205, filed on June 3, 2019, entitled “PHASE ESTIMATION WITH RANDOMIZED HAMILTONIANS,” which claims the benefit of U.S. Provisional Application No. 62 / 819,301, filed on March 15, 2019, entitled “PHASE ESTIMATION WITH RANDOMIZED HAMILTONIANS,” both of which are incorporated herein by reference in their entirety. Technical Field

[0003] The present application relates generally to quantum computing. In more detail, example methods are disclosed in which Hamiltonian terms are randomized and the accuracy of the random draw approximation is adapted as the required accuracy in phase estimation increases. Summary of the invention

[0004] Existing methods for simulating the dynamics of physical systems use either deterministic or randomly selected Hamiltonian terms. In the present disclosure, example methods are disclosed in which the Hamiltonian terms are randomized and the accuracy of the random draw approximation is adapted as the accuracy required in the phase estimate increases. This reduces the number of quantum gates required and, in some cases, reduces the number of qubits used in the simulation.

[0005] Embodiments include randomizing the phase estimate by replacing the Hamiltonian with a randomly generated Hamiltonian at each simulation. Additional embodiments relate to using randomization within an iterative phase estimation algorithm to select Hamiltonian terms for inclusion in the approximation and their ordering. Certain embodiments relate to using an importance function based on the importance of a term in the ground state to determine whether it is included in the randomly sampled Hamiltonian. Additional embodiments relate to using importance sampling based on a variational approximation to a ground state (such as, but not limited to, a CISD state). Certain embodiments relate to using an adaptive Bayesian approach in conjunction with this process to quantify the accuracy of the Hamiltonian required given the current uncertainty in the eigenvalues ​​being estimated by the algorithm.

[0006] In this application, an example method for performing quantum simulation uses adaptive Hamiltonian randomization. The specific embodiments described should not be interpreted as limiting, as the disclosed method actions can be performed individually, in different orders, or at least partially simultaneously with each other. In addition, any disclosed method or method action can be performed together with any other method or method action disclosed herein. In a specific embodiment, a Hamiltonian to be calculated by a quantum computer device is input; randomization is used within a phase estimation algorithm to reduce the number of Hamiltonian terms in the Hamiltonian; and a quantum circuit description for the Hamiltonian with a reduced number of Hamiltonian terms is output.

[0007] In some embodiments, the reduction includes selecting one or more random Hamiltonian terms based on an importance function; reweighting the selected random Hamiltonian terms based on the importance of each of the selected Hamiltonian random terms; and generating a quantum circuit description using the reweighted random terms. Some embodiments also include implementing a quantum circuit described by the quantum circuit description in a quantum computing device; and measuring the quantum state of the quantum circuit. Additional embodiments include re-performing the method based on results from the measurement (e.g., using an iterative process). In some embodiments, the iterative process includes calculating an expected accuracy value for the Hamiltonian; calculating a standard deviation for the Hamiltonian based on the results of the implementation and the measurement; and comparing the expected accuracy value to the standard deviation. Some embodiments also include changing the order of the Hamiltonian terms based on the reduction. Some embodiments also include applying an importance function to terms of the Hamiltonian of the ground state; and selecting one or more random Hamiltonian terms based at least in part on the importance function. Some embodiments include using importance sampling based on a variational approximation to the ground state. Some embodiments also include quantifying the accuracy required for the Hamiltonian using an adaptive Bayesian method given an estimate of the current uncertainty in the eigenvalues.

[0008] Other embodiments include one or more computer-readable media storing computer-executable instructions that, when executed by a computer, cause the computer to perform a method comprising inputting a Hamiltonian to be calculated by a quantum computer device; using randomization within a phase estimation algorithm to reduce the number of Hamiltonian terms in the Hamiltonian; and outputting a quantum circuit description for the Hamiltonian with the reduced number of Hamiltonian terms.

[0009] The foregoing and other objects, features and advantages of the disclosed technology will become more apparent from the following detailed description made with reference to the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] Figure 1 A quantum circuit for performing iterative phase estimation is shown.

[0011] Figures 2 to 9Included are graphs showing the average ground state energy change (compared to the unsampled Hamiltonian), the ground state energy variance on the sampled Hamiltonian, the average qubit requirement, and the average number of terms in the sampled Hamiltonian for Li2 (as a function of the number of samples used to generate the Hamiltonian and the value of the parameter p).

[0012] Fig.10 is a flow chart illustrating an example method for implementing an importance sampling simulation method according to an embodiment of the disclosed technology.

[0013] Fig.11 is a flow chart illustrating an example method for performing quantum simulations using adaptive Hamiltonian randomization.

[0014] Fig.12 A generalized example of a suitable classical computing environment is shown in which aspects of the described embodiments may be implemented.

[0015] Fig.13 Examples of possible network topologies (eg, client-server networks) for implementing systems in accordance with the disclosed technology are shown.

[0016] Fig.14 Another example of a possible network topology (eg, a distributed computing environment) for implementing a system in accordance with the disclosed technology is shown.

[0017] Fig.15 An exemplary system for implementing the disclosed technology is shown.

[0018] Fig.16 is a flow chart illustrating an example method for performing quantum simulations using adaptive Hamiltonian randomization. DETAILED DESCRIPTION

[0019] I. Introduction

[0020] Not all Hamiltonian terms are created equal in quantum simulations. Naturally occurring Hamiltonians from chemistry, materials, and other applications often consist of terms that are so small that they are negligible. These terms are often removed from the Hamiltonian before it reaches the simulator. Other terms that formally appear in the Hamiltonian are removed not because of their norm, but because they are not expected to affect the quantity of interest. For example, in quantum chemistry one often chooses the active orbital space and ignores any orbitals outside the active space. This results in many large terms being omitted from the Hamiltonian.

[0021] This process (often called decimation) typically involves systematically removing terms from the Hamiltonian and simulating the dynamics. The idea behind this approach is to remove terms from the Hamiltonian until the maximum allowed change in the eigenvalues ​​is comparable to the desired level of accuracy. For chemistry, chemical accuracy sets a natural accuracy threshold for such simulations, but this accuracy requirement does not generally need to be considered a constant.

[0022] One of the example innovations of the present disclosure is that in iterative phase estimation, the number of terms used in the Hamiltonian should not ideally remain constant. The reason is that when one is trying to learn a given bit of, for example, the binary expansion of the eigenphase, the higher order bits are mostly irrelevant. Simulations of much lower precision can be tolerated than when learning the higher order bits. Then, as the iterative phase estimation proceeds through the bits of the phase estimate, it makes sense to adapt the number of terms in the Hamiltonian. Example embodiments of the disclosed technology provide a systematic method for removing terms and provide a formal proof that such a process need not significantly affect the result of the phase estimate or its probability of success.

[0023] One of the concepts behind the example high-level decimation procedure is to estimate a priori which terms in the Hamiltonian are important, using a form of importance sampling. These randomized Hamiltonians are then used within the simulation circuit to prepare approximate ground states. It is thus shown, using an analysis reminiscent of the Zeno effect or the quantum adiabatic theorem, that the errors in the eigenstates prepared in each round of phase estimation need not have a substantial impact on the posterior mean of the eigenphases estimated from the true Hamiltonian. This suggests that, with appropriate assumptions about the eigenvalue gap, the procedure can be used to reduce the time complexity of the simulation, and in some cases even the space complexity, by identifying qubits that are not needed for the level of accuracy required for the simulation.

[0024] The present disclosure first reviews iterative phase estimation and Bayesian inference, which are used to quantify the maximum error in phase inference. The present disclosure then goes on to examine the impact of using a random Hamiltonian on the eigenphase produced by phase estimation in a simple case, where a fixed but random Hamiltonian is used at each step of the iterative phase estimation. The more complex case is then examined, where the iterative phase estimation circuit -iHt Each repetition of is implemented using a different random Hamiltonian. Theoretical analysis concludes that if the eigenvalue difference of the original Hamiltonian is large enough, the success probability will not be significantly reduced. In addition, a numerical example of this sampling process is shown, from which it can be concluded that the example sampling process of the Hamiltonian can have a significant impact on the number of terms in the Hamiltonian, and even in some cases, on the number of qubits used in the simulation.

[0025] II. Iterative Phase Estimation

[0026] The idea behind iterative phase estimation is based on constructing a quantum circuit as the target of an interferometer, where the unitary one one wishes to detect is applied to one of the two branches of the interferometer but not to the other. When the quantum state is allowed to interfere with itself at the end of the protocol, the interference pattern reveals the eigenphase. This process allows estimating the eigenvalues ​​of U within the standard quantum limit (i.e., the number of applications of U required to estimate the phase within an error ∈ is θ(1 / ∈ 2 ). If the quantum state is allowed to pass repeatedly through an interferometer circuit (or an entangled input is used), this scaling can be reduced to θ(1 / ∈), known as the Heisenberg limit. Figure 1 In particular, Figure 1 A quantum circuit for performing iterative phase estimation is shown. M is the number of repetitions of the unitary U (not necessarily an integer), and θ is the phase change between the auxiliary |0> and |1> states.

[0027] In U|ψ>=e iφ |ψ>, the phase estimation circuit is easy to analyze. If U is repeated M times and θ is the phase change, then using these parameters Figure 1 The likelihood of a given measurement o∈{0,1} of the circuit in is

[0028]

[0029] Many free parameters can be used when designing iterative phase estimation experiments. In particular, the rules used to generate M and θ for each experiment vary fundamentally with the method used to process the data returned by these experiments. Methods such as the Kitaev phase estimation algorithm, robust phase estimation, information theoretic phase estimation, or any number of approximate Bayesian methods provide good heuristics for choosing these parameters. In this disclosure, it is assumed that one does not wish to specify any of these methods for selecting experiments, nor does one wish to focus on the specific data processing methods used. Nevertheless, a Bayesian approach is relied upon to discuss the effects of the randomized Hamiltonian on the eigenphase estimate.

[0030] Bayes' theorem can be interpreted as providing the correct method for updating beliefs about some facts given a set of experimental evidence and a prior belief. The experimenter's initial belief is encoded by a prior distribution Pr(φ). In many cases it is appropriate to set Pr(φ) to a uniform distribution on [0, 2π) to represent a state of maximum ignorance about the eigenphase. However, in quantum simulations, if each step in the phase estimation uses And for different t j obey A wider range of priors can then be chosen, since such experiments can learn E0, rather than experiments with fixed t, which would yield φ = E0t mod 2π.

[0031] Therefore, Bayes' theorem gives that the posterior distribution Pr(φ|o; φ, M) should be

[0032]

[0033] Given a complete dataset instead of a single data point, we can get

[0034]

[0035] This probability distribution encodes the entire knowledge state of the experimenter, assuming that the data are processed optimally.

[0036] It is not customary to return the posterior distribution (or an approximation thereof) as an output of a phase estimation protocol. Instead, a point estimate of φ is given. The most commonly used estimate is the maximum a posteriori (MAP) estimate, which is simply the φ with the greatest probability. While this quantity has a nice operational interpretation, for the purposes of this disclosure it suffers from a number of deficiencies. The main drawback here is that the MAP estimate is not robust, in that if two different values ​​of φ have comparable likelihoods, then a small error in the likelihood can result in a radical change in the MAP estimate. For this purpose, the posterior mean is a better estimator for this purpose, which is formally The posterior mean can be viewed as an estimate of the mean squared error in any unbiased estimate of φ, and is therefore well motivated. It also has the property of being robust to small perturbations in the likelihood, a feature used below to estimate the impact on the experimental results of the phase estimation.

[0037] III. Error in the Likelihood Function

[0038] A. Unitary Linear Combinations

[0039] Linear combinations of unitary methods for quantum simulation have rapidly become a popular approach for simulating Hamiltonian dynamics in quantum systems. Unlike the Trotter decomposition, many of these methods do not necessarily produce unitary approximations that simulate quantum dynamics. This means that it is not possible to directly use Stone's theorem to argue that linear combinations of unitary methods can replace the e -iHt To achieve In turn, the standard analysis of error propagation from the Trotter decomposition to the estimation stage in iterative phase estimation fails, since one cannot directly reason about The characteristic value of .

[0040] Here, to partially address this issue, the impact that such errors may have on the likelihood function of iterative phase estimation is discussed.

[0041] Lemma 1. Let V be a possible non-unitary operation that can be non-deterministically performed by a quantum computer in a controlled manner, such that there exists a unitary U with ||UV||≤δ<1. If the likelihood function defined on V is but

[0042] Proof. Assume o = 0. Then, for the input state |ψ>, the error of the likelihood function output by the iterative phase estimation is

[0043]

[0044] Since P(0|Et;M,θ)+P(1|Et;M,θ)=1, it follows that the same bounds must also apply for o=1. Therefore, the result applies to any O claimed.

[0045] This result is straightforward but important because it allows the maximum error in the mean of the posterior distribution to be propagated through the iterative phase estimation protocol. The ability to propagate these errors will ultimately allow us to show that iterative phase estimation can be used to estimate eigenvalues ​​from linear combinations of unitary methods.

[0046] B. Undersampled Hamiltonian

[0047] Now consider the case of uniformly sampling terms from the Hamiltonian. Let the Hamiltonian be L simulatable Hamiltonians H l The sum of Always, consider the eigenstates of H |ψ> and their corresponding eigenenergy E. According to the original quantities, by transforming the term By uniformly sampling, a new Hamiltonian can be constructed

[0048]

[0049] When the Hamiltonian is randomly undersampled, errors are naturally introduced. The main question is not how big such errors are, but how they affect the iterative phase estimation protocol. The following lemma shows that the effect on the likelihood function can be arbitrarily small.

[0050] Lemma 2. Let l i is a family of indices that map {1, ..., m} → {1, ..., L} independently of replacement by uniformly sampling elements from {1, ..., L}, and let {H l : l = 1, ..., L} is The corresponding Hamiltonian family. For |ψ>, the eigenstates of H make H|ψ>=E|ψ> and Corresponding to the eigenstate H samp |ψ i >=E i |ψ i>, then the error in the likelihood function of the phase estimate is in the limit of large m. samp The above disappears with high probability:

[0051]

[0052] Proof. Because is uniformly sampled, so each group of items {l i} may be equal, and through the expected linear Know

[0053] The second moment is easily computed from the independence of the distribution.

[0054]

[0055] Due to different sequences i is chosen uniformly at random, so the results come from observing

[0056] According to the third-order perturbation theory, the leading-order change of any eigenvalue is within the error range The inner is O(<ψ|(HH samp )|ψ>). Therefore, according to equation (6), the variance of this change is

[0057]

[0058] This further indicates that the perturbed eigenstate |ψ i > has eigenvalues

[0059]

[0060] According to Markov inequality, its probability on i is high. Therefore, according to Taylor's theorem and equation (1),

[0061]

[0062] Its probability on i is high.

[0063] This result shows that if the coefficients of the Hamiltonian to be included in the undersampled Hamiltonian are uniformly sampled, the error in the estimate of the Hamiltonian can be made arbitrarily small. In this case, taking m → ∞ does not cause the simulation cost to diverge (as it does in many sampling problems). This is because once every possible term is included in the Hamiltonian, there is no point in undersampling, and H can also be reduced to samp Take H to eliminate the variance of the likelihood function caused by undersampling the Hamiltonian. In general, in order to ensure that the error in the likelihood function is at most a constant, it is necessary to take This therefore shows that, as with any iterative phase estimation algorithm, (barring problems with accidentally excited states due to perturbations) a good estimate of the eigenphase is obtained by adjusting m to be inversely proportional to M squared.

[0064] IV. Bayesian Phase Estimation Using Random Hamiltonian

[0065] Theorem 3. Let E be an event, let P(E|θ) and P′(E|θ) be two likelihood functions for θ∈[-π,π), such that max θ (|P(E|θ)-P′(E|θ)|)≤△, and further assume that for the prior P(θ), min(P(E), P′(E))≥2△. We get

[0066]

[0067] And if Where 1-|P′(E j |θ) / P(E j |θ)|≤γ, then

[0068] |∫θ(P(θ|E)-P′(θ|E))dθ|≤5π((1+γ) N -1).

[0069] Proof. According to the triangle inequality, we get

[0070] |P(E)-P′(E)|=|∫P(θ)(P(E|θ)-P′(E|θ))dθ|≤△

[0071] (10) Therefore, from the assumption that P′(E) ≥ 2△, we can conclude that

[0072]

[0073] Therefore, we get

[0074]

[0075] Now, if we assume that the likelihood function can be decomposed into N experiments, we can get

[0076]

[0077] According to the triangle inequality

[0078]

[0079] Solving this recursive relation yields

[0080]

[0081] Therefore, the results are as follows.

[0082] V. Variation of the Posterior Mean Using a Random Hamiltonian

[0083] In this section, we analyze the changes in the posterior mean of the estimated phase, assuming that the joint likelihood of all experiments undergoes a random variation of δ(φ).

[0084]

[0085] here, is the experimental parameter given the real phase φ and the original Hamiltonian and A series of N results when The joint likelihood of . is the joint likelihood of using a new random Hamiltonian in each experiment. equal vector, replicates are for each experiment performed in the series; M i is the number of repetitions in the ith experiment.

[0086] First, we can work backwards from the assumption that the joint likelihood changes by some amount δ(φ) to determine an upper bound on the acceptable difference in the ground state energies between the true and random Hamiltonians. We can do this by working backwards from the change in the joint likelihood of all experiments, to the change in the likelihoods of the individual experiments, to the corresponding tolerable difference between the ground state energies. Second, we can use this result to determine the change in the posterior mean with respect to the energy difference, and its standard deviation over a collection of randomly generated Hamiltonians.

[0087] A. Changes in Joint Likelihood

[0088] The random Hamiltonian for each experiment results in a random change in the joint likelihood of a sequence of outcomes.

[0089]

[0090] Suppose one wants to determine the maximum possible change in the posterior mean under this change in likelihood. The mean change in the likelihood under the assumed prior can be at most The posterior is

[0091]

[0092] Progress can be made in constraining the variation in the posterior by first constraining the variation in the joint likelihood according to the variation in the likelihoods of the individual experiments, as follows.

[0093] Lemma 4. Let P(o j |φ;M j ,θ j ) is the result of the jth experiment of Hamiltonian Hj The likelihood of P′(o j |φ;M j ,θ j )=P(o j |φ;M j ,θ j )+∈ j (φ) is the likelihood of the randomly generated Hamiltonian Hj. Assume that for all experiments j, Nmax j (|∈ j (φ)| / P(o j |φ,M j ,θ j ))<1 and |∈ j (φ)|≤P(o j |φ,M j ,θ j ) / 2. Therefore, the average change in the joint likelihood across all N experiments is,

[0094]

[0095] At most

[0096] Proof. We can use the likelihood P′(o j |φ;M j ,θ j )=P(o j |φ;M j ,θ j )+∈ j Change of (φ)∈ j (φ) to write the joint likelihood. The joint likelihood is so

[0097]

[0098] This gives the ratio of the changed to the unchanged joint likelihood,

[0099]

[0100] For example, this can be linearized and simplified using inequalities for logarithms and exponentials. By solving inequalities for the upper or lower bounds of these two functions, we can and P(o j |φ;M j ,θ j ), and the change in the likelihood of a single experiment ∈ j (φ) to determine the upper limit of |δ(φ)|.

[0101] The inequality that can be used to clamp the ratio is

[0102] For x≤0, 1-|x|≤exp(x)

[0103] For x<1, exp(x)≤1+2|x|

[0104] For |x|≤1 / 2, -2|x|≤log(1+x)

[0105] against log(1+x)≤|x| (twenty one)

[0107] In order for all four inequalities to hold, we must have,

[0108] N max j (|∈ j (φ)| / P(o j |φ,M j ,θ j ))<1 (for the exponential inequality), and for all j (for the logarithmic inequality), |∈ j (φ)|≤P(o j |φ,M j ,θ j ) / 2. Use them to determine an upper bound on the ratio of the changed to the unchanged likelihood,

[0109]

[0110] On the other hand, use them to determine the lower limit of the ratio,

[0111]

[0112] The upper and lower limits are identical before the sign. This allows one to combine them directly, obtaining

[0113]

[0114] From this, we can find an upper limit on the mean change on the posterior by using the triangle inequality:

[0115] Therefore, the change in the joint likelihood is bounded according to the change in the likelihood of the individual experiments. These results allow bounding the change in the posterior mean according to the change in the likelihood of the individual experiments ∈j(φ).

[0116] B. Changes in the Posterior Mean

[0117] You can use the assumption to define the change in the posterior mean.

[0118] Lemma 5. Assume that in addition to the assumptions in Lemma 4 Besides, the difference between the posterior mean seen with the ideal likelihood and the perturbed likelihood is at most

[0119]

[0120] Proof. The bound on the difference between posterior means can be solved by bounding the pointwise difference between the variational posterior and the posterior using the original Hamiltonian,

[0121]

[0122] As a first step, we can change the posterior The denominator of is capped:

[0123]

[0124] In both inequalities, the assumption is used Using this, again use The pointwise difference between the posterior points is at most

[0125]

[0126] With this, we can define the change in the posterior mean,

[0127]

[0128] Now, the bound of Lemma 4 allows to define the likelihood of individual experiments ∈ j The change of (φ) is used to define the change of the posterior mean,

[0129]

[0130] where in the last step, you can multiply and divide by This is

[0131]

[0132] C. Acceptable changes in phase

[0133] Another question is what are the bounds on the change in the posterior mean with respect to phase changes.

[0134] Theorem 6. If for all j and x∈[-π,π) For each of the N experiments, assuming Lemma 5, the eigenphase {φ′ j : j = 1, ... N} and the eigenphase φ of the true Hamiltonian obeys |φ-φ′ j |≤△φ and P(oj |φ,M j ,θ j )∈θ(1), then the change in the posterior mean of the eigenphase due to the inaccuracy of the eigenvalue of the intervening Hamiltonian follows

[0135]

[0136] Furthermore, if for all j, ∑ j M j ∈O(1 / ∈ φ ) and P(o j |φ;M j ,θ j )∈θ(1), then

[0137]

[0138] Proof. We can define ∈ according to the change ∆φ of the phase applied to the ground state j (φ) represents the change in the posterior mean. Recall that the likelihood of the random Hamiltonian is

[0139] P′(o j |φ;M j ,θ j )=P(o j |φ;M j ,θ j )+∈ j (φ)

[0140] , (32) where the unchanged likelihood of the jth experiment is Therefore, using the upper bound of the derivative sin(x)≤|x|,

[0141]

[0142] In summary, the error in the posterior mean is at most

[0143]

[0144] As a result, if the branch [-π, π) is chosen, the absolute value of the posterior mean is at most π

[0145] VI. Changes in the eigenphase using a new random Hamiltonian at each repetition

[0146] The variance of the applied phase can be reduced by generating a different Hamiltonian in each repetition. However, this is not without cost: it can be seen as leading to a failure probability in the evolution or more generally to an additional phase shift.

[0147] The reason for this reduction in variance is somewhat complicated to formalize mathematically, and it arises when computing the variance of |∆φ|. Rather than just a single Hamiltonian exponential, the variance is in M j Hamiltonian. Therefore, it simply expands to Instead of expanding to (φ est The trade-off is that by reducing the variance of the phase in this way, additional changes in the phase are introduced. If one does not do this across multiple steps, one will always be adiabatically bounded with the same error Hamiltonian. Instead, this means that one is only approximately adiabatic, at the expense of reducing the variance by a factor M j . Since additional changes are also Linear, this can lead to improvements. Usually the gap is required to be small, and λ is required j ∝||H j -H j-1 ||Smaller.

[0148] Then, the competition between standard deviations is or And this new change in M j So depending on the gap, it might be hard to get a tight bound to show that this is better, and one should not just stick to the higher variance from a single Hamiltonian.

[0149] A. Failure probability of the algorithm

[0150] For phase estimation, the variance of phase estimation can be reduced by randomizing the repetitions of each experiment. j The jth experiment of the repetition (recall that M j Not necessarily an integer), one is divided into Repeat.

[0151] In each repetition, a new Hamiltonian H can be randomly generated k Each Hamiltonian H k The ground state and energy of are slightly different from those of the other Hamiltonians.

[0152] The reason for reducing the estimated phase variance is that the phases between repetitions are uncorrelated: whereas for the single Hamiltonian case, the phase exp(-iMφ est ) has a variance of When simulating a different random Hamiltonian in each repetition (and estimating the sum of the phases ), the variance is

[0153] By evolving under different random instantiations of the Hamiltonian in each repetition, the variance of the phase is reduced quadratically; the only cost is that the algorithm now has a failure probability (leaving the ground state from repetition to repetition, e.g., in the case of k-1 The ground state to H k The first case is easier to analyze: as Lemma 7 shows, the failure probability can be arbitrarily small as long as the gap is small enough. This can be done by considering the success probability of the algorithm as the sum of the true sum of the ground state energies. This is done by measuring the probability of remaining in the ground state in a sequence of random Hamiltonians. In the second case, if the state leaves the ground space only for short time intervals during the evolution, a bound on the difference between the eigenvalues ​​can be proved in Lemma 8.

[0154] Lemma 7. Consider the Hamiltonian sequence, M>1. Let γ be the minimum difference between the ground energy and the first excitation energy of any Hamiltonian, Similarly, let λ = max k ||H k -H k-1 || is the maximum difference between any two Hamiltonians in the sequence. M The probability of leaving the ground state when transferring sequentially is at most 0<∈<1, provided that

[0155]

[0156] Prove. is the Hamiltonian H k The i-th eigenstate of is the corresponding energy. Given that the algorithm starts with The probability of remaining in the ground state in all M steps is

[0157]

[0158] This is the probability that the algorithm remains in the ground state in each segment. We establish bounds to simplify this expression. Let λ k V k =H k -H k-1 , where λ can be chosen k Make ||V k ||=1 to simplify the proof. k V k Considered as H k-1 The disturbance on k-1 The component of the change in the ground state is given by the derivative

[0159]

[0160] Multiply by λ=max|λ k |Bounded, where the maximization is over k and the perturbation for a given k.

[0161] Use this,

[0162]

[0163] This gets in make where we can again choose κ k So that |φ k > is normalized,

[0164]

[0165] Can be based on Solution because Since for x∈[0,1], so

[0166]

[0167] because Finally, back to Because κ k ≤1 (this is true because ||V k ||=1), the difference between the ground states of the two Hamiltonians is at most

[0168]

[0169] This means that the probability of overlap between the ground states of any two adjacent Hamiltonians is Across M segments (M-1 transitions), the success probability is at least If you want the maximum failure probability to be fixed at 0<∈<1, you need

[0170]

[0171] If only the ground state |ψ0〉 of the original Hamiltonian can be prepared, then the success probability has an additional factor In this case, Lemma 7 can be applied and ||H-H1|| is included in the maximization of λ. In addition, since Where E1-E0 is the gap between the ground state and the first excited state of H, then

[0172]

[0173] If this happens, then each Hamiltonian remains in the ground state with probability 1-∈ throughout the simulation. In this case, the total accumulated phase is

[0174] in

[0175] B. Phase shift due to Hamiltonian error

[0176] The analysis of the phase difference can be summarized by determining the difference between the expected (adiabatic) unitary and the true unitary. Evolving sequentially under M random Hamiltonians, , is applied to each new Hamiltonian H k The unitary

[0177]

[0178] The adiabatic unit used in the ideal case is

[0179]

[0180] The difference between the two is that H k The real time evolution of U k For H k The eigenstate of the applied phase, and the adiabatic unitary U k，ad Apply the eigenphase and then change H k Each eigenstate of H is mapped to k+1 This means that if the system starts from the ground state H1, the sequence The phase that will be applied to it is proportional to the sum of the ground state energies of each Hamiltonian in the sequence. In contrast, will include the Hamiltonians H k The contributions of many different eigenstates.

[0181] Unitary k with U k，ad The differences between are defined as follows.

[0182] Lemma 8. Let To H k The projector on the ground state of , and let the assumption of Lemma 7 hold.

[0183] and The difference between the eigenvalues ​​of (where ∆t is the simulation time) is at most

[0184]

[0185] First, you can use the identity The next Hamiltonian Hk+1 The eigenstate of

[0186]

[0187] make and (When p = 1). In a sense, the new eigenstate Written as and state A slight change, this is the choice Using this definition, we can continue to simplify U k :

[0188]

[0189] Now the boundaries can be well defined Note: k，ad Exactly equal to U k 1 in the first sum of

[0190]

[0191] Define || U k -U k，ad The final step is to define Similar to how to define δ pl For p≠l, △ pl Given by the following formula

[0192]

[0193] So, similar to |δ0| in Lemma 7, 2 and∑ l＞0 |δ l | 2 The same boundaries, The upper limit of is given by

[0194]

[0195] This completes the proof.

[0196] Theorem 9. Consider the Hamiltonian sequence, M>1. Let γ be the minimum difference between the ground energy and the first excitation energy of any Hamiltonian, Similarly, let λ = max k ||H k -H k-1 || is the maximum difference between any two Hamiltonians in the sequence. The maximum error of the estimated unitary eigenphase obtained by multiplying these M Hamiltonians is

[0197]

[0198] The probability of failure is at most ∈, provided that

[0199]

[0200] Proof. Lemma 8 gives and Across the entire sequence, we have

[0201]

[0202] This is the maximum possible difference between the accumulated phases of the ideal and actual sequences, assuming that the system leaves the ground state for at most one repetition at a time.

[0203] The probability of leaving the ground state as part of a Landau Zener process caused by a measurement at an adjacent value of the Hamiltonian is the probability of failure ∈ occurring at each projection under the assumption of Lemma 7, if

[0204]

[0205] So, the result can be easily derived from these two results.

[0206] VII. Importance Sampling

[0207] The basic idea behind the example method of extracting the Hamiltonian is importance sampling. This method has been used a lot in merging, but it can be used slightly differently here. The idea behind importance sampling is to reduce the variance of the mean of the quantity by reweighting the sum. In particular, the mean of the N numbers F(j) can be written as

[0208]

[0209] where f(j) is the importance of a given term. This suggests that the initial unweighted average can be viewed as a reweighted amount x j / (f(j)N). Although this is true for x j It has no effect on the mean of , and it can significantly reduce the sample variance of the mean, so it is widely used in statistics to provide a more accurate estimate of the mean. The best importance function to use in these cases is f(j)∝|x j |. In this case, it is straightforward to see that if x j is a constant, then the variance of the resulting distribution is effectively zero. In this case, a straightforward forward calculation shows that the optimal variance is

[0210]

[0211] If the sign of the number is constant, then the optimal variance in (54) is actually zero. While this may seem surprising, it becomes less mysterious when one notes that in order to compute the optimal importance function one needs to estimate the ensemble mean. In most cases this would defeat the purpose of importance sampling. Therefore, if one wants to gain advantage from importance sampling for Hamiltonian simulations, it is important to show that one can use it even with inexact importance functions, e.g., ones that can be computed efficiently using classical computers.

[0212] It will now be shown below how this robustness is maintained.

[0213] Lemma 10. Let is an unknown probability distribution from which one can sample, and let is a known function, so that for all j, where |δ j |≤|F(j)| / 2. If the importance sampling and importance function When used together, the variance follows

[0214]

[0215] Proof. The proof is a straightforward exercise in the triangle inequality, once |δ j |≤|F(j)| / 2 and 1 / (1-|x|)≤1+2|x| for all x∈[-1 / 2,1 / 2]:

[0216]

[0217] This bound is tight because as the max k |δ k |→0, the upper bound of the variance converges to the best achievable variance

[0218] In applications such as quantum chemistry simulations, what one wants to do is to minimize the variance in Lemma 2. This minimum variance can be obtained by choosing f(j)∝|<ψ|H j |ψ>|. However, the task of computing such a function is at least as difficult as solving the desired eigenvalue estimation problem. The natural approach to take is to draw inspiration from Lemma 10, rather than adopting in is a valid computable ansatz state, such as a CISD state. However, in practice, the importance of a given term may not be fully predicted by the ansatz prediction. In this case, a hedging strategy can be used, where for some ρ∈[0,1], f(j)∝(1-ρ)(H j >+ρ||Hj ||. This strategy allows to smoothly interpolate between the importance represented by the magnitude of the Hamiltonian term and the expected value of the surrogate in the ground state.

[0219] VIII. Numerical results

[0220] The present disclosure has shown that iterative phase estimation can be used using randomized Hamiltonians. To illustrate the effectiveness of the example embodiments, consider two examples of diatomic molecules: dilithium and hydrogen chloride. In both cases, the molecules are prepared on a minimal STO6G basis and use CISD states, which are found by variably minimizing the ground state energy over all states within a 2 excitation range away from the Hartree Fock state. The Hamiltonian terms can then be randomly sampled and several quantities of interest can be examined, including the average ground state energy, the variance of the ground state energy, and the average number of terms in the Hamiltonian. Interestingly, one can also look at the number of qubits present in the model. This may vary because some randomly sampled Hamiltonians will actually select terms in the Hamiltonian that are not coupled to the rest of the system. In these cases, the number of qubits required to represent the state may actually be lower than the total number typically expected.

[0221] from Figures 2 to 5 and Figures 6 to 9 It can be seen that the estimate of the ground state energy changes radically with the degree of hedging used. It is found that if ρ = 1 in both cases, the variance of the ground state energy is very large in all cases, as expected since importance sampling has little effect in this case. In contrast, it is found that when ρ = U, the importance of the Hamiltonian terms from the CISD state is maximally prioritized, which leads to very parsimonious models, but for even 10 7 The ground state energy of a randomly selected term (some of which may be repeated) changes by 1 Ha. If we use moderate hedging (p = 2 × 10 -5 ), note that the change in the ground state energy is minimized, assuming that 10% of the chemical accuracy or an energy change of 0.1 mHa is acceptable for Hamiltonian truncation errors. For dilithium, this represents a 30% reduction in the number of terms in the Hamiltonian; for HC 1, this reduces the number of terms in the Hamiltonian by a factor of 3. Since the cost of Trotter-Suzuki chemistry simulations scales superlinearly with the number of terms in the Hamiltonian, this constitutes a significant reduction in complexity.

[0222] It can also be noted that for the case of dilithium, the number of qubits required to perform the simulation varies from run to run. In contrast, chlorine does not exhibit this behavior. This difference stems from the fact that dilithium requires 6 electrons to be located in 20 spin orbits. Hydrogen chloride consists of 18 electrons and the example Fock space also consists of 20 spin orbits. Therefore, almost every spin orbit is associated with this, which explains why the number of spin orbits required to express dilithium to a certain accuracy varies, while it does not for HC 1. This illustrates that embodiments of the disclosed randomization process can be used to help select the active space for simulation as the accuracy required for the Hamiltonian increases during the phase estimation process.

[0223] Figures 2 to 9 Included are graphs 200, 300, 400, 500, 600, 700, 800, and 900 showing the average ground state energy change (compared to the unsampled Hamiltonian), the ground state energy variance on the sampled Hamiltonian, the average qubit requirement, and the average number of terms in the sampled Hamiltonian for Li2 as a function of the number of samples used to generate the Hamiltonian and the value of the parameter p. Hamiltonian H α One of the items has probability p α ∝(1-ρ) <H α >+ρ||H α || is sampled, where the expected value is taken in the CISD state.

[0224] IX. Example Embodiments

[0225] In this section, example methods for performing the disclosed techniques are disclosed. The specific embodiments described should not be construed as limiting, as the disclosed method actions can be performed individually, in different orders, or at least partially simultaneously with each other. In addition, any disclosed method or method action can be performed together with any other method or method action disclosed herein.

[0226] Fig.10 is a flowchart 1000 illustrating an example method for implementing an importance sampling simulation method according to an embodiment of the disclosed technology.

[0227] Fig.11 is a flow chart 1100 illustrating an example method for performing quantum simulations using adaptive Hamiltonian randomization.

[0228] Fig.16 is a flow chart 1600 illustrating an example method for performing quantum simulations using adaptive Hamiltonian randomization.

[0229] At 1610, a Hamiltonian to be calculated by a quantum computer device is input.

[0230] At 1612, randomization is used within the phase estimation algorithm to reduce the number of Hamiltonian terms in the Hamiltonian.

[0231] At 1614, a quantum circuit description of the Hamiltonian with a reduced number of Hamiltonian terms is output.

[0232] In some embodiments, the reduction includes selecting one or more random Hamiltonian terms based on an importance function; reweighting the selected random Hamiltonian terms based on the importance of each of the selected Hamiltonian random terms; and generating a quantum circuit description using the reweighted random terms. Some embodiments also include implementing a quantum circuit described by the quantum circuit description in a quantum computing device; and measuring the quantum state of the quantum circuit. Additional embodiments include re-performing the method based on results from the measurement (e.g., using an iterative process). In some embodiments, the iterative process includes calculating an expected accuracy value for the Hamiltonian; calculating a standard deviation for the Hamiltonian based on the results of the implementation and the measurement; and comparing the expected accuracy value to the standard deviation. Some embodiments also include changing the order of the Hamiltonian terms based on the reduction. Some embodiments also include applying an importance function to terms of the Hamiltonian of the ground state; and selecting one or more random Hamiltonian terms based at least in part on the importance function. Some embodiments include using importance sampling based on a variational approximation to the ground state. Some embodiments also include quantifying the accuracy required for the Hamiltonian using an adaptive Bayesian method given an estimate of the current uncertainty in the eigenvalues.

[0233] Other embodiments include one or more computer-readable media storing computer-executable instructions that, when executed by a computer, cause the computer to perform a method comprising inputting a Hamiltonian to be calculated by a quantum computer device; using randomization within a phase estimation algorithm to reduce the number of Hamiltonian terms in the Hamiltonian; and outputting a quantum circuit description for the Hamiltonian with the reduced number of Hamiltonian terms.

[0234] The method may include selecting one or more random Hamiltonian terms based on an importance function; reweighting the selected random Hamiltonian terms based on the importance of each of the selected Hamiltonian random terms; and generating a quantum circuit description using the reweighted random terms. The method may also include implementing a quantum circuit described by the quantum circuit description in a quantum computing device; and measuring a quantum state of the quantum circuit. The method may also include calculating an expected accuracy value for the Hamiltonian; calculating a standard deviation for the Hamiltonian based on the results of the implementation and the measurement; comparing the expected accuracy value to the standard deviation; and re-performing the reduction based on the results of the comparison.

[0235] Another embodiment is a system comprising a quantum computing system; and a classical computing system configured to communicate with and control the quantum computing system. In such an embodiment, the classical computing system is further configured to: input a Hamiltonian to be calculated by a quantum computer device; use randomization within an iterative phase estimation algorithm to reduce the number of Hamiltonian terms in the Hamiltonian; and output a quantum circuit description for the Hamiltonian with a reduced number of Hamiltonian terms. The classical computing system may also be configured to: select one or more random Hamiltonian terms based on an importance function; reweight the selected random Hamiltonian terms based on the importance of each of the selected Hamiltonian random terms; and generate a quantum circuit description using the reweighted random terms. The classical computing system may also be configured to cause a quantum circuit described by the quantum circuit description to be implemented by a quantum computing device; and measure the quantum state of the quantum circuit. Further, the classical computing system may also be configured to calculate an expected accuracy value for the Hamiltonian; calculate a standard deviation for the Hamiltonian based on the results of the implementation and measurement; compare the expected accuracy value with the standard deviation; and re-execute the reduction based on the result of the comparison. In further embodiments, the configuration of the classical computing system may also be configured such that one or more unnecessary qubits are omitted as part of the randomization.

[0236] X. Example Computing Environment

[0237] Fig.12 A generalized example of a suitable classical computing environment 1200 in which aspects of the described embodiments may be implemented is shown. The computing environment 1200 is not intended to suggest any limitation as to the scope of use or functionality of the disclosed technology, as the techniques and tools described herein may be implemented in a variety of general-purpose or special-purpose environments with computing hardware.

[0238] refer to Fig.12 , the computing environment 1200 includes at least one processing device 1210 and a memory 1220. Fig.12 , this most basic configuration 1230 is included within the dashed line. A processing device 1210 (e.g., a CPU or microprocessor) executes computer executable instructions. In a multi-processing system, multiple processing devices execute computer executable instructions to increase processing power. Memory 1220 can be volatile memory (e.g., registers, cache, RAM, DRAM, SRAM), non-volatile memory (e.g., ROM, EEPROM, flash memory), or some combination of the two. Memory 1220 stores software 1280 implementing tools for performing any disclosed technique for operating a quantum computer to perform Hamiltonian randomization as described herein.

[0239] The memory 1220 may also store software 1280 for synthesizing, generating, or compiling quantum circuits for performing any of the disclosed techniques.

[0240] The computing environment may have additional features. For example, the computing environment 1200 includes storage 1240, one or more input devices 1250, one or more output devices 1260, and one or more communication connections 1270. An interconnection mechanism (not shown) (such as a bus, controller, or network) interconnects the components of the computing environment 1200. Typically, operating system software (not shown) provides an operating environment for other software executed in the computing environment 1200 and coordinates the activities of the components of the computing environment 1200.

[0241] The storage device 1240 may be removable or non-removable and include one or more disks (e.g., hard drives), solid-state drives (e.g., flash drives), magnetic tapes or cartridges, CD-ROMs, DVDs, or any other tangible, non-volatile storage medium that can be used to store information and that can be accessed within the computing environment 1200. The storage device 1240 may also store instructions for software 1280 to implement any of the disclosed techniques. The storage device 1240 may also store instructions for software 1280 to generate and / or synthesize any of the described techniques, systems, or quantum circuits.

[0242] The input device(s) 1250 may be a touch input device such as a keyboard, a touch screen, a mouse, a pen, a trackball, a voice input device, a scanning device, or another device that provides input to the computing environment 1200. The output device(s) 1260 may be a display device (e.g., a computer monitor, a laptop display, a smartphone display, a tablet display, a netbook display, or a touch screen), a printer, a speaker, or another device that provides output from the computing environment 1200.

[0243] Communication connection(s) 1270 enable communication with another computing entity via a communication medium. The communication medium transmits information such as computer executable instructions or other data in a modulated data signal. A modulated data signal is a signal that has one or more of its characteristics set or changed to encode the information in the signal. By way of example and not limitation, communication media include wired or wireless technologies implemented using electrical, optical, RF, infrared, acoustic or other carriers.

[0244] As described above, various methods and techniques for performing Hamiltonian randomization, for controlling a quantum computing device to perform circuit design or compilation / synthesis as disclosed herein, can be described in the general context of computer-readable instructions stored on one or more computer-readable media. A computer-readable medium is any available medium (e.g., a memory or storage device) that can be accessed within or by a computing environment. A computer-readable medium includes a tangible computer-readable memory or storage device, such as memory 1220 and / or storage device 1240, and does not include a propagating carrier wave or signal itself (a tangible computer-readable memory or storage device does not include a propagating carrier wave or signal itself).

[0245] Various embodiments of the methods disclosed herein may also be described in the general context of computer-executable instructions (e.g., those included in program modules) executed by a processor in a computing environment. Typically, program modules include routines, programs, libraries, objects, classes, components, data structures, etc. that perform specific tasks or implement specific abstract data types. In various embodiments, the functionality of program modules may be combined or split between program modules as needed. The computer-executable instructions of program modules may be executed in a local or distributed computing environment.

[0246] Fig.13 An example of a possible network topology 1300 (e.g., a client-server network) for implementing a system according to the disclosed technology is depicted in FIG. 1300 . A networked computing device 1320 may be, for example, a computer running a browser or other software connected to a network 1312 . The computing device 1320 may have a computer system such as a server computer or a server computer. Fig.12The computer architecture shown and discussed above. The computing device 1320 is not limited to a traditional personal computer, but may include other computing hardware (e.g., a smart phone, a laptop, a tablet computer, or other mobile computing devices, servers, network devices, special devices, etc.) configured to be connected to and communicate with the network 1312. In addition, the computing device 1320 may include an FPGA or other programmable logic device. In the illustrated embodiment, the computing device 1320 is configured to communicate with the computing device 1330 (e.g., a remote server, such as a server in a cloud computing environment) via the network 1312. In the illustrated embodiment, the computing device 1320 is configured to transmit input data to the computing device 1330, and the computing device 1330 is configured to implement circuit generation / compilation / synthesis techniques for controlling the quantum computing device to perform any disclosed embodiment and / or for generating quantum circuits for performing any technology disclosed herein. The computing device 1330 can output results to the computing device 1320. Any data received from the computing device 1330 can be stored or displayed on the computing device 1320 (e.g., displayed as data on a graphical user interface or web page at the computing device 1320). In the illustrated embodiment, the illustrated network 1312 may be implemented as a local area network (LAN) using wired networking (e.g., Ethernet IEEE Standard 802.3 or other appropriate standards) or wireless networking (e.g., IEEE Standards 802.11a, 802.11b, 802.11g or 802.11h or one of other appropriate standards). Alternatively, at least a portion of the network 1312 may be the Internet or a similar public network and operate using an appropriate protocol (e.g., HTTP protocol).

[0247] Fig.14 Another example of a possible network topology 1400 (e.g., a distributed computing environment) for implementing a system according to the disclosed technology is depicted in FIG. A networked computing device 1420 may be, for example, a computer running a browser or other software connected to the network 1112. The computing device 1420 may have a plurality of components such as a processor, a processor, and a processor. Fig.12The computer architecture shown and discussed above. In the illustrated embodiment, the computing device 1420 is configured to communicate with multiple computing devices 1430, 1431, 1432 (e.g., remote servers or other distributed computing devices, such as one or more servers in a cloud computing environment) via a network 1412. In the illustrated embodiment, each of the computing devices 1430, 1431, 1432 in the computing environment 1400 is used to perform at least a portion of the Hamiltonian randomization technique and / or for controlling a quantum computing device to perform at least a portion of the technique of any disclosed embodiment and / or for generating a circuit generation / compilation / synthesis technique for a quantum circuit for performing any technique disclosed herein. In other words, the computing devices 1430, 1431, 1432 form a distributed computing environment in which aspects of the technique for performing any technique disclosed herein and / or quantum circuit generation / compilation / synthesis processes are shared across multiple computing devices. The computing device 1420 is configured to transmit input data to computing devices 1430, 1431, 1432, which are configured to implement processes such as distributed, including the execution of any disclosed method or the creation of any disclosed circuit, and provide results to the computing device 1420. Any data received from the computing devices 1430, 1431, 1432 can be stored or displayed on the computing device 1420 (for example, displayed as data on a graphical user interface or web page at the computing device 1420). The illustrated network 1412 can be the network described above. Fig.13 Any network discussed.

[0248] refer to Fig.15 , an exemplary system for implementing the disclosed technology includes a computing environment 1500. In the computing environment 1500, a compiled quantum computer circuit description (including a quantum circuit for performing any disclosed technology as disclosed herein) can be used to program (or configure) one or more quantum processing units so that the quantum processing units implement the circuit described by the quantum computer circuit description.

[0249] Environment 1500 includes one or more quantum processors 1502 and one or more readout devices 1508. The quantum processing unit executes a quantum circuit precompiled and described by a quantum computer circuit description. The quantum processing unit can be one or more of the following, but is not limited to: (a) a superconducting quantum computer; (b) an ion trap quantum computer; (c) a fault-tolerant architecture for quantum computing; and / or (d) a topological quantum architecture (e.g., a topological quantum computing device using Majorana zero modes). The precompiled quantum circuit (including any disclosed circuit) can be sent to (or otherwise applied to) (multiple) quantum processing units via control lines 1506 under the control of a quantum processor controller 1520. The quantum processor controller (QP controller) 1520 can be connected to a classical processor 1510 (e.g., having a circuit as described above). Fig.12 The QP controller 1520 and the QP sub-controller 1504 operate together to implement the desired quantum computing process. In the example shown, the QP controller 1520 also implements the desired quantum computing process via one or more QP sub-controllers 1504, which are particularly suitable for controlling a corresponding one of the (multiple) quantum processors 1502. For example, in one example, the quantum controller 1520 facilitates the implementation of the compiled quantum circuit by sending instructions to one or more memories (e.g., cryogenic memories), and then passing the instructions to (multiple) cryogenic control units (e.g., (multiple) QP sub-controllers 1504), for example, the cryogenic control unit transmits a pulse sequence representing a gate to the (multiple) quantum processor 1502 for implementation. In other examples, the QP controller 1520 and the QP sub-controller 1504 operate to provide appropriate magnetic retention, encoding operations, or other such control signals to the (multiple) quantum processors to implement the operations described by the compiled quantum computer circuit. The (multiple) quantum controllers may also interact with the readout device 1508 to help control and implement the desired quantum computing process (e.g., once available, by reading or measuring data results from the quantum processing unit, etc.)

[0250] refer to Fig.15 , compilation is the process of converting a high-level description of a quantum algorithm into a quantum computer circuit description, the circuit description comprising a sequence of quantum operations or gates, which may include circuits disclosed herein (e.g., circuits configured to perform one or more processes as disclosed herein). Compilation may be performed by a compiler 1522 using a classical processor 1510 of the environment 1500 (e.g., Fig.12 ), the processor 1510 loads the high-level description from the memory or storage device 1512 and stores the generated quantum computer circuit description in the memory or storage device 1512.

[0251] In other embodiments, the compilation and / or verification may be performed by a remote computer 1560 (e.g., a computer having a processor as described above with respect to Fig.12 The remote computer 1560 may be remotely executed by the remote computer 1560 of the computing environment described above, and the remote computer 1560 stores the resulting quantum computer circuit description in one or more memories or storage devices 1562 and transmits the quantum computer circuit description to the computing environment 1500 for implementation in the (multiple) quantum processors 1502. In addition, the remote computer 1500 may store the high-level description in a memory or storage device 1562 and transmit the high-level description to the computing environment 1500 for compilation and use with the (multiple) quantum processors. In any of these scenarios, the results of the computations performed by the (multiple) quantum processors may be transmitted to the remote computer after and / or during the computation process. Further, the remote computer may communicate with the (multiple) QP controllers 1520 so that the quantum computation process (including any compilation, verification, and QP control processes) may be remotely controlled by the remote computer 1560. In general, the remote computer 1560 communicates with the (multiple) QP controllers 1520, the compiler 1522 / synthesizer, and / or the verification tool 1523 via the communication connection 1550.

[0252] In certain embodiments, environment 1500 may be a cloud computing environment that provides quantum processing resources of environment 1500 to one or more remote computers (such as remote computer 1560) over a suitable network (which may include the Internet).

[0253] XI. Conclusion

[0254] This application shows that iterative phase estimation is more flexible than might have been previously believed, and that the number of terms in the Hamiltonian is randomized at each step of the iterative phase estimation without substantially affecting the underlying variance of the unbiased estimator of the eigenphase. It is further shown numerically that by using this strategy of subsampling the Hamiltonian terms, simulations can be performed with fewer Hamiltonian terms than required by conventional methods. These reductions in the number of terms have a direct impact on the complexity of Trotter-Suzuki based simulations, and indirectly on quantization and truncated Taylor series simulation methods, as they also reduce the 1-norm of the Hamiltonian term vector.

[0255] The principles of the disclosed technology have been described and shown with reference to the illustrated embodiments, and it should be appreciated that the illustrated embodiments can be modified in arrangement and detail without departing from these principles. For example, the elements of the illustrated embodiments shown in software can be implemented in hardware, and vice versa. In addition, the technology from any example can be combined with the technology described in any one or more other examples. It should be understood that processes and functions such as those described with reference to the illustrated examples can be implemented in a single hardware or software module, or a separate module can be provided. The above-mentioned specific arrangement is provided for convenience of explanation, and other arrangements may also be used.

Claims

1. A method of operating a quantum computing device, comprising: inputting a Hamiltonian to be calculated by the quantum computing device; using randomization within a phase estimation algorithm to reduce the number of Hamiltonian terms in the Hamiltonian; as well as A quantum circuit description of the Hamiltonian with a reduced number of Hamiltonian terms is output.

2. The method of claim 1, wherein the method is performed by a classical computer.

3. The method of claim 1, wherein the reducing comprises: selecting one or more random Hamiltonian terms based on an importance function; reweighting the selected random Hamiltonian terms based on the importance of each of the selected random Hamiltonian terms; The quantum circuit description is generated using the reweighted random Hamiltonian terms.

4. The method according to claim 3, further comprising: implementing a quantum circuit described by the quantum circuit description in the quantum computing device; A quantum state of the quantum circuit is measured.

5. The method of claim 4, further comprising re-performing the method of claim 4 based on results from the measuring. The method of claim 5 , wherein the re-executing is performed based on an iterative process.

7. The method of claim 6, wherein the iterative process comprises: calculating a desired precision value for the Hamiltonian; calculating a standard deviation for the Hamiltonian based on the results of the implementation and the measurement; as well as The expected accuracy value is compared to the standard deviation.

8. The method of claim 1, further comprising changing an order of the Hamiltonian term based on the reduction.

9. The method according to claim 1, further comprising: applying an importance function to terms of the Hamiltonian in the ground state; as well as One or more random Hamiltonian terms are selected based at least in part on the importance function.

10. The method according to claim 1, further comprising: Use importance sampling based on a variational approximation to the ground state.

11. The method according to claim 1, further comprising: Given an estimate of the current uncertainty in the eigenvalues, an adaptive Bayesian approach is used to quantify the required accuracy for the Hamiltonian.

12. One or more computer-readable media storing computer-executable instructions, which, when executed by a computer, cause the computer to perform a method comprising: Input the Hamiltonian to be calculated by the quantum computer device; using randomization within a phase estimation algorithm to reduce the number of Hamiltonian terms in the Hamiltonian; as well as A quantum circuit description of the Hamiltonian with a reduced number of Hamiltonian terms is output.

13. One or more computer readable media according to claim 12, wherein the method further comprises: selecting one or more random Hamiltonian terms based on an importance function; reweighting the selected random Hamiltonian terms based on the importance of each of the selected random Hamiltonian terms; as well as The quantum circuit description is generated using the reweighted random Hamiltonian terms.

14. One or more computer readable media according to claim 13, wherein the method further comprises: causing the quantum circuit described by the quantum circuit description to be implemented by a quantum computing device; as well as A quantum state of the quantum circuit is measured.

15. The one or more computer-readable media of claim 14, wherein the method further comprises: calculating a desired precision value for the Hamiltonian; calculating a standard deviation for the Hamiltonian based on the results of the implementation and the measurement; comparing the expected accuracy value to the standard deviation; and The reducing is re-executed based on the result of the comparing.

16. A system for operating a quantum computing device, comprising: Quantum computing systems; as well as A classical computing system configured to communicate with and control the quantum computing system, the classical computing system further configured to: inputting a Hamiltonian to be calculated by the quantum computing system; using randomization within a phase estimation algorithm to reduce the number of Hamiltonian terms in the Hamiltonian; as well as A quantum circuit description of the Hamiltonian with a reduced number of Hamiltonian terms is output.

17. The system of claim 16, wherein the classical computing system is further configured to: selecting one or more random Hamiltonian terms based on an importance function; reweighting the selected random Hamiltonian terms based on the importance of each of the selected random Hamiltonian terms; and The quantum circuit description is generated using the reweighted random Hamiltonian terms.

18. The system of claim 17, wherein the classical computing system is further configured to: causing the quantum circuit described by the quantum circuit description to be implemented by the quantum computing device; and A quantum state of the quantum circuit is measured.

19. The system of claim 18, wherein the classical computing system is further configured to: calculating a desired precision value for the Hamiltonian; calculating a standard deviation for the Hamiltonian based on the results of the implementation and the measurement; comparing the expected accuracy value to the standard deviation; and The reducing is re-executed based on the result of the comparing.

20. The system of claim 16, wherein the classical computing system is further configured to omit one or more unnecessary qubits based on the reduced number of Hamiltonian terms.

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