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arXiv:1712.05384

Simulation of low-depth quantum circuits as complex undirected graphical models

Sergio Boixo, Sergei V. Isakov, Vadim Smelyanskiy, Hartmut Neven

Published Dec 14, 2017• 73 citations
arXivHTMLPDF

Abstract

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 $5 \times 9$ qubits of depth 37, $7 \times 8$ qubits of depth 27, and $10 \times (κ> 10)$) 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 $7 \times 7$ qubits and depth 40 remain currently out of reach. Follow up work on a supercomputer environment will tighten this bound.

DOI
10.48550/arxiv.1712.05384

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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.
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With the rapid progress made by industry and academia, quantum computers with dozens of qubits or even larger size are being realized. However, the fidelity of existing quantum computers often sharply decreases as the circuit depth increases. Thus, an ideal quantum circuit simulator on classical computers, especially on high-performance computers, is needed for benchmarking and validation. We design a large-scale simulator of universal random quantum circuits, often called “quantum supremacy circuits”, and implement it on Sunway TaihuLight. The simulator can be used to accomplish the following two tasks: 1) Computing a complete output state-vector; 2) Calculating one or a few amplitudes. We target the simulation of 49-qubit circuits. For task 1), we successfully simulate such a circuit of depth 39, and for task 2) we reach the 55-depth level. To the best of our knowledge, both of the simulation results reach the largest depth for 49-qubit quantum supremacy circuits.
As Moore's law reaches its limits, quantum computers are emerging with the promise of dramatically outperforming classical computers. We have witnessed the advent of quantum processors with over 505050 quantum bits (qubits), which are expected to be beyond the reach of classical simulation. Quantum supremacy is the event at which the old Extended Church-Turing Thesis is overturned: A quantum computer performs a task that is practically impossible for any classical (super)computer. The demonstration requires both a solid theoretical guarantee and an experimental realization. The lead candidate is Random Circuit Sampling (RCS), which is the task of sampling from the output distribution of random quantum circuits. Google recently announced a 53−53-53−qubit experimental demonstration of RCS. Soon after, classical algorithms appeared that challenge the supremacy of random circuits by estimating their outputs. How hard is it to classically simulate the output of random quantum circuits? We prove that estimating the output probabilities of random quantum circuits is formidably hard (#P\#P#P-Hard) for any classical computer. This makes RCS the strongest candidate for demonstrating quantum supremacy relative to all other proposals. The robustness to the estimation error that we prove may serve as a new hardness criterion for the performance of classical algorithms. To achieve this, we introduce the Cayley path interpolation between any two gates of a quantum computation and convolve recent advances in quantum complexity and information with probability and random matrices. Furthermore, we apply algebraic geometry to generalize the well-known Berlekamp-Welch algorithm that is widely used in coding theory and cryptography. Our results imply that there is an exponential hardness barrier for the classical simulation of most quantum circuits.
A central question of quantum computing is determining the source of the advantage of quantum computation over classical computation.Even though simulating quantum dynamics on a classical computer is thought to require exponential overhead in the worst case, efficient simulations are known to exist in several special cases.It was widely assumed that these easy-to-simulate cases as well as any yetundiscovered ones could be avoided by choosing a quantum circuit at random.We prove that this intuition is false by showing that certain families of constant-depth, 2D random circuits can be approximately simulated on a classical computer in time only linear in the number of qubits and gates, even though the same families are capable of universal quantum computation and are hard to exactly simulate in the worst case (under standard hardness assumptions).While our proof applies to specific random circuit families, we demonstrate numerically that typical instances of more general families of sufficiently shallow constantdepth 2D random circuits are also efficiently simulable.We propose two classical simulation algorithms.One is based on first simulating spatially local regions which are then "stitched" together via recovery maps.The other reduces the 2D simulation problem to a problem of simulating a form of 1D dynamics consisting of alternating rounds of random local unitaries and weak measurements.Similar processes have recently been the subject of an intensive research focus, which has observed that the dynamics generally undergo a phase transition from a low-entanglement (and efficient-to-simulate) regime to a highentanglement (and inefficient-to-simulate) regime as measurement strength is varied.Via a mapping from random quantum circuits to classical statistical mechanical models, we give analytical evidence that a similar computational phase transition occurs for both of our algorithms as parameters of the circuit architecture like the local Hilbert space dimension and circuit depth are varied and, additionally, that the effective 1D dynamics corresponding to sufficiently shallow random quantum circuits falls within the efficient-to-simulate regime.Implementing the latter algorithm for the depth-3 "brickwork" architecture, for which exact simulation is hard, we find that a laptop could simulate typical instances on a 409 × 409 grid with a total variation distance error less than 0.01 in approximately one minute per sample, a task intractable for previously known circuit simulation algorithms.Numerical results support our analytic evidence that the algorithm is asymptotically efficient.
Empirical evidence for a gap between the computational powers of classical and quantum computers has been provided by experiments that sample the output distributions of two-dimensional quantum circuits. Many attempts to close this gap have utilized classical simulations based on tensor network techniques, and their limitations shed light on the improvements to quantum hardware required to frustrate classical simulability. In particular, quantum computers having in excess of approximately 50 qubits are primarily vulnerable to classical simulation due to restrictions on their gate fidelity and their connectivity, the latter determining how many gates are required (and, therefore, how much infidelity is suffered) in generating highly entangled states. Here, we describe recent hardware upgrades to Quantinuum’s H2 quantum computer, enabling it to operate on up to 56 qubits with arbitrary connectivity and 99.843(5)% two-qubit gate fidelity. We define a class of circuits with random geometries that become hard to classically simulate in very low depth and implement them utilizing the flexible connectivity of H2. A careful analysis demonstrating the fast saturation of classical simulation complexity with depth indicates that H2 can yield data well beyond the reach of state-of-the art classical simulation methods at unprecedented fidelities. We find that the considerable difficulty of classically simulating H2 is likely limited only by qubit number, demonstrating the promise and scalability of the quantum charge-coupled device architecture as continued progress is made toward building larger machines.
As quantum computers improve in the number of qubits and fidelity, the question of when they surpass state-of-the-art classical computation for a well-defined computational task is attracting much attention. The leading candidate task for this milestone entails sampling from the output distribution defined by a random quantum circuit. We develop a massively-parallel simulation tool Rollright that does not require inter-process communication (IPC) or proprietary hardware. We also develop two ways to trade circuit fidelity for computational speedups, so as to match the fidelity of a given quantum computer --- a task previously thought impossible. We report massive speedups for the sampling task over prior software from Microsoft, IBM, Alibaba and Google, as well as supercomputer and GPU-based simulations. By using publicly available Google Cloud Computing, we price such simulations and enable comparisons by total cost across hardware platforms. We simulate approximate sampling from the output of a circuit with 7x8 qubits and depth 1+40+1 by producing one million bitstring probabilities with fidelity 0.5%, at an estimated cost of $35184. The simulation costs scale linearly with fidelity, and using this scaling we estimate that extending circuit depth to 1+48+1 increases costs to one million dollars. Scaling the simulation to 10M bitstring probabilities needed for sampling 1M bitstrings helps comparing simulation to quantum computers. We describe refinements in benchmarks that slow down leading simulators, halving the circuit depth that can be simulated within the same time.