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arXiv:1710.09364

Recursive Path-Summing Simulation of Quantum Computation

Andrew Shi

Published Oct 25, 2017• 1 citation
arXivHTMLPDF

Abstract

Classical simulation of quantum computation has often been viewed as the method to determine where the horizon of quantum supremacy is located---that is, where quantum computation can no longer be simulated by classical methods. As of now, the 50 qubit threshold for quantum supremacy has been determined largely by the state vector simulation method's exponential space demands placing an upper bound on simulation memory capabilities. To investigate this claim, we present and test an implementation of a known path integral simulation algorithm running in linear space; the method is based on recursively traversing the underlying computation tree for quantum algorithms and summing over possible amplitudes. We find that the implementation is able to simulate the hidden subgroup method (HSP) standard method---a notable class of circuits including Shor's algorithm amongst others---in a reasonable amount of time using extremely low memory, as well as other circuits with similar parameters. The performance results of this algorithm suggest that it can serve as a feasible alternative to state vector simulation and that with respect to the HSP standard method, quantum supremacy may be more accurately measured using the recursive path-summing method on large numbers of qubits, compared to the state vector method.

DOI
10.48550/arxiv.1710.09364

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Classical simulation of quantum computation has often been viewed as the method to determine where the horizon of quantum supremacy is located---that is, where quantum computation can no longer be simulated by classical methods. As of now, the 50 qubit threshold for quantum supremacy has been determined largely by the state vector simulation method's exponential space demands placing an upper bound on simulation memory capabilities. To investigate this claim, we present and test an implementation of a known path integral simulation algorithm running in linear space; the method is based on recursively traversing the underlying computation tree for quantum algorithms and summing over possible amplitudes. We find that the implementation is able to simulate the hidden subgroup method (HSP) standard method---a notable class of circuits including Shor's algorithm amongst others---in a reasonable amount of time using extremely low memory, as well as other circuits with similar parameters. The performance results of this algorithm suggest that it can serve as a feasible alternative to state vector simulation and that with respect to the HSP standard method, quantum supremacy may be more accurately measured using the recursive path-summing method on large numbers of qubits, compared to the state vector method.
We propose an approximate classical algorithm for estimating quantum circuits' transition amplitudes, ⟨Ψf∣U^∣Ψ0⟩\langle Ψ_f | \hat{U} | Ψ_0 \rangle⟨Ψf​∣U^∣Ψ0​⟩, based on Monte Carlo sums over bidirectionally sampled Feynman paths. The algorithm generates Feynman paths, i.e., sequences of intermediate computational basis states, by combining information from random walks initiated at both the input and output states, Ψ0Ψ_0Ψ0​ and ΨfΨ_fΨf​, respectively. It further combines this bidirectional approach with Multiple Importance Sampling, a powerful variance reduction technique that avoids variance peaks caused by divisions by the very low sampling probabilities associated with hard to reach paths. Empirical data demonstrate the superior convergence rate exhibited by the bidirectional algorithm over a unidirectional, forward-only sampling approach. This superiority is due to the effectiveness of the proposed algorithm in finding and weighting paths with non-zero amplitude in an extremely sparse path space. Sum of paths algorithms exhibit very favourable space complexity. Monte Carlo replaces exhaustive enumeration of the exponentially large path space by stochastic sampling, the computational cost being determined by the variance of the estimator. This work demonstrates these two features by presenting results on very large quantum circuits, with up to 4096 qubits.
We propose an approximate classical algorithm for estimating quantum circuits' transition amplitudes, ⟨Ψf∣U^∣Ψ0⟩\langle Ψ_f | \hat{U} | Ψ_0 \rangle⟨Ψf​∣U^∣Ψ0​⟩, based on Monte Carlo sums over bidirectionally sampled Feynman paths. The algorithm generates Feynman paths, i.e., sequences of intermediate computational basis states, by combining information from random walks initiated at both the input and output states, Ψ0Ψ_0Ψ0​ and ΨfΨ_fΨf​, respectively. It further combines this bidirectional approach with Multiple Importance Sampling, a powerful variance reduction technique that avoids variance peaks caused by divisions by the very low sampling probabilities associated with hard to reach paths. Empirical data demonstrate the superior convergence rate exhibited by the bidirectional algorithm over a unidirectional, forward-only sampling approach. This superiority is due to the effectiveness of the proposed algorithm in finding and weighting paths with non-zero amplitude in an extremely sparse path space. Sum of paths algorithms exhibit very favourable space complexity. Monte Carlo replaces exhaustive enumeration of the exponentially large path space by stochastic sampling, the computational cost being determined by the variance of the estimator. This work demonstrates these two features by presenting results on very large quantum circuits, with up to 4096 qubits.
Quantum computation was envisioned by Feynman as a valuable means of solving quantum problems [1]. The question is, how do we prove the superiority of quantum computing over classical devices? A common misconception in describing the power of a quantum computer over a classical computer is that quantum bits can be prepared in a superposition of an exponential number of states, which cannot be achieved with classical bits. This statement is certainly true, but it does not tell us, specifically, what problems quantum computers can efficiently solve while classical computers fail to do so. In fact, based exactly on the technique of Feynman's path integral, the output (i.e. transition amplitude) of any quantum circuit can be calculated classically with a polynomial amount of memory (although the temporal cost would be exponential). Therefore any quantum computation can be simulated classically; the key question is about the efficiency. Much effort has been made trying to pin down the power of quantum computers based on the machinery of the computational complexity theory. In particular, the class of computational (decision) problems that can be solved efficiently by a quantum computer is called BQP (bounded-error quantum polynomial time); the counterpart for classical computers is called P (polynomial time). Of course, BQP cannot be less than P; in principle quantum computers can simulate classical computers efficiently. The point is that the foundation of quantum computation cannot be established without a proof showing that BQP is strictly larger than P (i.e. BQP ≠ P). In this sense, this question is as interesting as the famous challenge of proving P ≠ NP (non-deterministic polynomial time). A well known instance of an NP problem is the factoring problem; Shor's quantum algorithm is capable of solving the factoring problem in polynomial time, which is not achievable with the best classical algorithm discovered so far. However, we still cannot rigorously exclude the possibility of the existence of an efficient classical algorithm for factoring. Although again without a proof, it is commonly believed that quantum computers are unable to solve some of the NP problems. In fact, the proof of this statement (if true) provides a possible path to prove P ≠ NP. A potentially less demanding question would be, when do we expect a quantum computer to be able to perform some well defined tasks (not necessarily related to any practical problem) that cannot be simulated with any currently available classical device, within a reasonable time? Once achieved, the status is called ‘quantum supremacy' [2]. One may ask, what about Grover's search algorithm, which provides a quadratic speedup over the classical search? Can we say that we can already achieve quantum supremacy with Grover's algorithm? The problem is that, instead of taking the large-N limit, quantum supremacy requires us to determine the actual number of qubits and gates that can no longer be simulated by classical computers within a reasonable time. To elaborate further, we remark that there can be two notions of classical simulation, namely ‘strong simulation' versus ‘weak simulation'. Strong simulation refers to the cases of calculating the transition probabilities, or expectation values of observables, to a high accuracy in polynomial time. Weak simulation requires the probability distributions to be accurately reproduced, which involves ‘sampling' the different outcomes of the quantum devices. Some quantum circuits that cannot be strongly simulated with an efficient classical means may have efficient classical methods for weak simulation [3]. However, we should be careful about the accuracy requirement in the simulation tasks. For example, for many decision problems, it may be sufficient to determine the transition amplitudes to within an additive error, instead of multiplicative error. In other words, the classical simulability of quantum computational problems depends on the accuracy requirement. Currently, there are three popular approaches for achieving quantum supremacy (see Fig. 1), namely (i) boson sampling [4], (ii) sampling IQP (instantaneous quantum polynomial) circuits [5], and (iii) sampling chaotic quantum circuits [6]. In all of these approaches, distributions of different bit strings or photon numbers are sampled from the quantum devices. Furthermore, they all involve the assumption that classical computers are unable to efficiently determine the transition amplitudes, and/or reproduce (or approximate) the distributions of quantum devices performing these tasks, in the sense of both strong and weak simulations. Three different approaches for achieving quantum supremacy: (a) boson sampling; (b) IQP circuits; (c) chaotic quantum circuits. In boson sampling, single photons are injected into different modes of a linear optical network. The task is to determine the photon distributions at the output ports. The key feature of boson sampling is that the transition amplitudes are related to the permanents of complex matrices, which are in general very difficult to calculate exactly, or approximate to within an multiplicative error; these problems belong to the #P-hard complexity class. Moreover, efficient weak simulation of boson sampling is believed to be impossible, unless the polynomial hierarchy collapses to its third level. However, a recent numerical study suggests that, in order to achieve quantum supremacy with boson sampling, one needs to simultaneously generate at least 50 or more single photons, which is still highly technologically challenging. An interesting question is whether linear optics, in the setting of boson sampling, can be applied to solve decision problems. In this case, the transition amplitudes may need to be determined to an additive error. However, we found that a large class of decision problems associated with boson sampling can be simulated by a classical computer [7], settling an open problem [8] by Aaronson and Hance. IQP circuits represent a simplified quantum-circuit model. The initial state starts from the all-zero state, |000…0〉, followed by applying Hadamard gates to each qubit. Then, diagonal gates are applied to the qubits, followed by applying Hadamard gates to each qubit again. The argument of showing the complexity in simulating IQP circuits is similar to that of boson sampling. In fact, the complexity of boson sampling was inspired by the complexity results of IQP circuits. However, when subject to noise, IQP circuits may become classically simulable [9]. Lastly, chaotic quantum circuits refer to the class of quantum circuits where two-qubit gates are applied regularly, while single-qubit gates are applied randomly from a gate set. The output distributions of these circuits approach the so-called Porter–Thomas distribution, which is a characteristic feature of quantum chaos. Recently, there have been several numerical investigations aiming to explore the ultimate limit of classical computing in simulating low-depth quantum circuits. For the ideal cases, one needs to consider both the qubit number and circuit depth for benchmarking quantum supremacy. However, in the presence of noise, these chaotic circuits may also become classically simulable for high-depth circuits [10]. Finally, other than these three approaches, one should expect that quantum supremacy can be achieved for many practical applications, e.g. simulation of quantum chemistry or quantum machine learning. When will it happen? Only time will tell!
The Feynman path integral formalism has inspired the development of memory-efficient and parallelizable classical algorithms for simulating quantum computers. We adapt this approach for the calculation of probability amplitudes of linear-optical boson sampling experiments, which involve Fock-state inputs, linear optical circuits, and photo-detection at the output. We describe this simulation method and compare it with alternative approaches. Additionally, we implement a linear-optical Feynman path simulator in open-source C code, enhancing its performance using tensor contraction techniques. Our method is benchmarked for low-depth linear optical circuits, where it offers advantages in runtime and memory efficiency.
Near term quantum computers with a high quantity (around 50) and quality (around 0.995 fidelity for two-qubit gates) of qubits will approximately sample from certain probability distributions beyond the capabilities of known classical algorithms on state-of-the-art computers, achieving the first milestone of so-called quantum supremacy. This has stimulated recent progress in classical algorithms to simulate quantum circuits. Classical simulations are also necessary to approximate the fidelity of multiqubit quantum computers using cross entropy benchmarking. Here we present numerical results of a classical simulation algorithm to sample universal random circuits, on a single workstation, with more qubits and depth than previously reported. For example, circuits with 5×95 \times 95×9 qubits of depth 37, 7×87 \times 87×8 qubits of depth 27, and 10×(κ>10)10 \times (κ> 10)10×(κ>10)) qubits of depth 19 are all easy to sample. We also show up to what depth the sampling, or estimation of observables, is trivially parallelizable. The algorithm is related to the "Feynmann path" method to simulate quantum circuits. For low-depth circuits, the algorithm scales exponentially in the depth times the smaller lateral dimension, or the treewidth, as explained in Boixo et. al., and therefore confirms the bounds in that paper. In particular, circuits with 7×77 \times 77×7 qubits and depth 40 remain currently out of reach. Follow up work on a supercomputer environment will tighten this bound.
Quantum performance simulators can provide practical metrics for the effectiveness of executing theoretical quantum information processing protocols on physical hardware. In this work we present a scheme to simulate the performance of fault tolerant quantum computation by automating the tracking of common fault paths for error propagation through a circuit and quantifying the fidelity of each qubit throughout the computation. Our simulation tool outputs the expected execution time, required number of qubits and the final error rate of running common fault tolerant protocols on a universal hardware, assumed to be a network of qubits with full connectivity. Our technique efficiently estimates the upper bound of error probability and provides a useful performance measure of the error threshold at low error rates where conventional Monte Carlo methods are ineffective. To verify the proposed simulator, we present simulation results comparing the execution of quantum adders which constitute a major part of Shor's algorithm.