Google's recent quantum supremacy experiment heralded a transition point where quantum computing performed a computational task, random circuit sampling, that is beyond the practical reach of modern supercomputers. We examine the constraints of the observed quantum runtime advantage in an extrapolation to circuits with a larger number of qubits and gates. Due to the exponential decrease of the experimental fidelity with the number of qubits and gates, we demonstrate for current fidelities a theoretical classical runtime advantage for circuits deeper than a few hundred gates, while quantum runtimes for cross-entropy benchmarking limit the region of a quantum advantage to a few hundred qubits. However, the quantum runtime advantage boundary in circuit width and depth grows exponentially with respect to reduced error rates, and our work highlights the importance of continued progress along this line. Extrapolations of measured error rates suggest that the limiting circuit size for which a computationally feasible quantum runtime advantage in cross-entropy benchmarking can be achieved approximately coincides with expectations for early implementations of the surface code and other quantum error correction methods. Thus the boundaries of quantum supremacy via random circuit sampling may fortuitously coincide with the advent of scalable, error corrected quantum computing in the near term.
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…
Although the emergence of a fully-functional quantum computer may still be far away from today, in the near future, it is possible to have medium-size, special-purpose, quantum devices that can perform computational tasks not efficiently simulable with any classical computer. This status is known as quantum supremacy (or quantum advantage), where one of the promising approaches is through the sampling of chaotic quantum circuits. Sampling of ideal chaotic quantum circuits has been argued to require an exponential time for classical devices. A major question is whether quantum supremacy can be …
Recent experiments completed by collaborating research groups from Google, NASA Ames, UC Santa Barbara, and others provided compelling evidence that quantum supremacy has finally been achieved on a superconducting quantum processor. The theoretical basis for these experiments depends on sampling the output distributions of random quantum circuits; unfortunately, understanding how this theoretical basis can be used to define quantum supremacy is an extremely difficult task. Anyone attempting to understand how this sampling task relates to quantum supremacy must study concepts from random matrix…
A critical question for the field of quantum computing in the near future is whether quantum devices without error correction can perform a well-defined computational task beyond the capabilities of state-of-the-art classical computers, achieving so-called quantum supremacy. We study the task of sampling from the output distributions of (pseudo-)random quantum circuits, a natural task for benchmarking quantum computers. Crucially, sampling this distribution classically requires a direct numerical simulation of the circuit, with computational cost exponential in the number of qubits. This requi…
Demonstrating quantum supremacy, a complexity-guaranteed quantum advantage against over the best classical algorithms by using less universal quantum devices, is an important near-term milestone for quantum information processing. Here we develop a threshold theorem for quantum supremacy with noisy quantum circuits in the pre-threshold region, where quantum error correction does not work directly. We show that, even in such a region, we can virtually simulate quantum error correction by postselection. This allows us to show that the output sampled from the noisy quantum circuits (without posts…
We propose a general tensor network method for simulating quantum circuits. The method is massively more efficient in computing a large number of correlated bitstring amplitudes and probabilities than existing methods. As an application, we study the sampling problem of Google's Sycamore circuits, which are believed to be beyond the reach of classical supercomputers and have been used to demonstrate quantum supremacy. Using our method, employing a small computational cluster containing 60 graphical processing units (GPUs), we have generated one million correlated bitstrings with some entries f…
It is believed that random quantum circuits are difficult to simulate classically. These have been used to demonstrate quantum supremacy: the execution of a computational task on a quantum computer that is infeasible for any classical computer. The task underlying the assertion of quantum supremacy by Arute et al. (Nature, 574, 505--510 (2019)) was initially estimated to require Summit, the world's most powerful supercomputer today, approximately 10,000 years. The same task was performed on the Sycamore quantum processor in only 200 seconds. In this work, we present a tensor network-based clas…
Compressible models extend the domain of simulable systems in quantum computers, but little is known about their precise limits of applicability. Using the theory of compressible matchgate circuits, we identify a class of quadratic fermionic Hamiltonians that can be simulated in compressed space. In particular, for systems of $n$ orbitals encoded to 2-local qubit models with nearest neighbour interactions, the ground state energy can be evaluated with $O\left(\log n\right)$ sets of measurements, independently of the number of dimensions in which the $n$ sites are arranged. We also provide an e…
It is known that several sub-universal quantum computing models, such as the IQP model, the Boson sampling model, the one-clean qubit model, and the random circuit model, cannot be classically simulated in polynomial time under certain conjectures in classical complexity theory. Recently, these results have been improved to "fine-grained" versions where even exponential-time classical simulations are excluded assuming certain classical fine-grained complexity conjectures. All these fine-grained results are, however, about the hardness of strong simulations or multiplicative-error sampling. It …
A critical milestone on the path to useful quantum computers is quantum supremacy - a demonstration of a quantum computation that is prohibitively hard for classical computers. A leading near-term candidate, put forth by the Google/UCSB team, is sampling from the probability distributions of randomly chosen quantum circuits, which we call Random Circuit Sampling (RCS). In this paper we study both the hardness and verification of RCS. While RCS was defined with experimental realization in mind, we show complexity theoretic evidence of hardness that is on par with the strongest theoretical propo…
We consider a model of quantum computation we call "Varying-$Z$" (V$Z$), defined by applying controllable $Z$-diagonal Hamiltonians in the presence of a uniform and constant external $X$-field, and prove that it is universal, even in 1D. Universality is demonstrated by construction of a universal gate set with $O(1)$ depth overhead. We then use this construction to describe a circuit whose output distribution cannot be classically simulated unless the polynomial hierarchy collapses, with the goal of providing a low-resource method of demonstrating quantum supremacy. The V$Z$ model can achieve …
The class of commuting quantum circuits known as IQP (instantaneous quantum polynomial-time) has been shown to be hard to simulate classically, assuming certain complexity-theoretic conjectures. Here we study the power of IQP circuits in the presence of physically motivated constraints. First, we show that there is a family of sparse IQP circuits that can be implemented on a square lattice of n qubits in depth O(sqrt(n) log n), and which is likely hard to simulate classically. Next, we show that, if an arbitrarily small constant amount of noise is applied to each qubit at the end of any IQP ci…
Quantum computing is of high interest because it promises to perform at least some kinds of computations much faster than classical computers. Arute et al. 2019 (informally, "the Google Quantum Team") report the results of experiments that purport to demonstrate "quantum supremacy" -- the claim that the performance of some quantum computers is better than that of classical computers on some problems. Do these results close the debate over quantum supremacy? We argue that they do not. We provide an overview of the Google Quantum Team's experiments, then identify some open questions in the quest…
Motivated by the recent experimental demonstrations of quantum supremacy, proving the hardness of the output of random quantum circuits is an imperative near term goal. We prove under the complexity theoretical assumption of the non-collapse of the polynomial hierarchy that approximating the output probabilities of random quantum circuits to within $\exp(-Ω(m\log m))$ additive error is hard for any classical computer, where $m$ is the number of gates in the quantum computation. More precisely, we show that the above problem is $\#\mathsf{P}$-hard under $\mathsf{BPP}^{\mathsf{NP}}$ reduction. I…
Instantaneous quantum polynomial time (IQP) is a model of (probably) non-universal quantum computation. Since it has been proven that IQP circuits are unlikely to be simulated classically up to a multiplicative error and an error in the $l_1$ norm, IQP is considered as one of the promising classes that demonstrates quantum supremacy. Although IQP circuits can be realized more easily than a universal quantum computer, demonstrating quantum supremacy is still difficult. It is therefore desired to find subclasses of IQP that are easy to implement. In this paper, by imposing some restrictions on I…
Recent advances on quantum computing hardware have pushed quantum computing to the verge of quantum supremacy. Random quantum circuits are outstanding candidates to demonstrate quantum supremacy, which could be implemented on a quantum device that supports nearest-neighbour gate operations on a two-dimensional configuration. Here we show that using the Projected Entangled-Pair States algorithm, a tool to study two-dimensional strongly interacting many-body quantum systems, we can realize an effective general-purpose simulator of quantum algorithms. This technique allows to quantify precisely t…
A major milestone in the era of noisy intermediate scale quantum computers is \textit{quantum supremacy} [Nature \textbf{574}, 505 (2019)] claimed on the Sycamore quantum processor of $53$ qubits, which can perform a random circuit sampling task within $200$ seconds while the same task is estimated to require a runtime of $10,000$ years on Summit. This record has been renewed with two recent experiments on the Zuchongzhi $2.0$ ($56$ qubits) and Zuchongzhi $2.1$ ($60$ qubits) quantum processors. On the other front of quantum supremacy comparison, there has also been continuous improvements on b…
We establish a connection between continuous-variable quantum computing and high-dimensional integration by showing that the outcome probabilities of continuous-variable instantaneous quantum polynomial (CV-IQP) circuits are given by integrals of oscillating functions in large dimensions. We prove two results related to the classical hardness of evaluating these integrals: (i) we show that there exist circuits such that these integrals are approximations of a weighted sum of #P-hard problems and (ii) we prove that calculating these integrals is as hard as calculating integrals of arbitrary bou…
Quantum computational supremacy arguments, which describe a way for a quantum computer to perform a task that cannot also be done by a classical computer, typically require some sort of computational assumption related to the limitations of classical computation. One common assumption is that the polynomial hierarchy (PH) does not collapse, a stronger version of the statement that P $\neq$ NP, which leads to the conclusion that any classical simulation of certain families of quantum circuits requires time scaling worse than any polynomial in the size of the circuits. However, the asymptotic na…
Results on the hardness of approximate sampling are seen as important stepping stones towards a convincing demonstration of the superior computational power of quantum devices. The most prominent suggestions for such experiments include boson sampling, IQP circuit sampling, and universal random circuit sampling. A key challenge for any such demonstration is to certify the correct implementation. For all these examples, and in fact for all sufficiently flat distributions, we show that any non-interactive certification from classical samples and a description of the target distribution requires …
We introduce a new open-source software library Jet, which uses task-based parallelism to obtain speed-ups in classical tensor-network simulations of quantum circuits. These speed-ups result from i) the increased parallelism introduced by mapping the tensor-network simulation to a task-based framework, ii) a novel method of reusing shared work between tensor-network contraction tasks, and iii) the concurrent contraction of tensor networks on all available hardware. We demonstrate the advantages of our method by benchmarking our code on several Sycamore-53 and Gaussian boson sampling (GBS) supr…
We introduce a distributed classical simulation algorithm for general quantum circuits, and present numerical results for calculating the output probabilities of universal random circuits. We find that we can simulate more qubits to greater depth than previously reported using the cluster supported by the Data Infrastructure and Search Technology Division of the Alibaba Group. For example, computing a single amplitude of an $8\times 8$ qubit circuit with depth $40$ was previously beyond the reach of supercomputers. Our algorithm can compute this within $2$ minutes using a small portion ($\appr…
We develop a high-performance tensor-based simulator for random quantum circuits(RQCs) on the new Sunway supercomputer. Our major innovations include: (1) a near-optimal slicing scheme, and a path-optimization strategy that considers both complexity and compute density; (2) a three-level parallelization scheme that scales to about 42 million cores; (3) a fused permutation and multiplication design that improves the compute efficiency for a wide range of tensor contraction scenarios; and (4) a mixed-precision scheme to further improve the performance. Our simulator effectively expands the scope…
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 $50$ 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…
While quantum speed-up in solving certain decision problems by a fault-tolerant universal quantum computer has been promised, a timely research interest includes how far one can reduce the resource requirement to demonstrate a provable advantage in quantum devices without demanding quantum error correction, which is crucial for prolonging the coherence time of qubits. We propose a model device made of locally-interacting multiple qubits, designed such that simultaneous single-qubit measurements on it can output probability distributions whose average-case sampling is classically intractable, u…
The last five years have seen a dramatic evolution of platforms for quantum computing, taking the field from physics experiments to quantum hardware and software engineering. Nevertheless, despite this progress of quantum processors, the field is still in the noisy intermediate-scale quantum (NISQ) regime, seriously limiting the performance of software applications. Key issues involve how to achieve quantum advantage in useful applications for quantum optimization and materials science, connected to the concept of quantum supremacy first demonstrated by Google in 2019. In this article we will …
Fine-grained quantum supremacy is a study of proving (nearly) tight time lower bounds for classical simulations of quantum computing under "fine-grained complexity" assumptions. We show that under conjectures on Orthogonal Vectors (OV), 3-SUM, All-Pairs Shortest Paths (APSP) and their variants, strong and weak classical simulations of quantum computing are impossible in certain exponential time with respect to the number of qubits. Those conjectures are widely used in classical fine-grained complexity theory in which polynomial time hardness is conjectured. All previous results of fine-grained…
One-parameter interpolations between any two unitary matrices (e.g., quantum gates) $U_1$ and $U_2$ along efficient paths contained in the unitary group are constructed. Motivated by applications, we propose the continuous unitary path $U(θ)$ obtained from the QR-factorization \[ U(θ)R(θ)=(1-θ)A+θB, \] where $U_1 R_1=A$ and $U_2 R_2=B$ are the QR-factorizations of $A$ and $B$, and $U(θ)$ is a unitary for all $θ$ with $U(0)=U_1$ and $U(1)=U_2$. The QR-algorithm is modified to, instead of $U(θ)$, output a matrix whose columns are proportional to the corresponding columns of $U(θ)$ and whose entr…
During the last ten years, superconducting circuits have passed from being interesting physical devices to becoming contenders for near-future useful and scalable quantum information processing (QIP). Advanced quantum simulation experiments have been shown with up to nine qubits, while a demonstration of Quantum Supremacy with fifty qubits is anticipated in just a few years. Quantum Supremacy means that the quantum system can no longer be simulated by the most powerful classical supercomputers. Integrated classical-quantum computing systems are already emerging that can be used for software de…
The notable claim of quantum supremacy presented by Google's team in 2019 consists of demonstrating the ability of a quantum circuit to generate, albeit with considerable noise, bitstrings from a distribution that is considered hard to simulate on classical computers. Verifying that the generated data is indeed from the claimed distribution and assessing the circuit's noise level and its fidelity is a purely statistical undertaking. The objective of this paper is to explain the relations between quantum computing and some of the statistical aspects involved in demonstrating quantum supremacy i…
Noise is the defining feature of the NISQ era, but it remains unclear if noisy quantum devices are capable of quantum speedups. Quantum supremacy experiments have been a major step forward, but gaps remain between the theory behind these experiments and their actual implementations. In this work we initiate the study of the complexity of quantum random circuit sampling experiments with realistic amounts of noise. Actual quantum supremacy experiments have high levels of uncorrected noise and exponentially decaying fidelities. It is natural to ask if there is any signal of exponential complexity…
The computational advantage of noisy quantum computers has been demonstrated by sampling the bitstrings of quantum random circuits. An important issue is how the performance of quantum devices could be quantified in the so-called "supremacy regime". The standard approach is through the linear cross entropy benchmark (XEB), where the theoretical value of the probability is required for each bitstring. However, the computational cost of XEB grows exponentially. So far, random circuits of the 53-qubit Sycamore chip were verified up to 10 cycles of gates only; the XEB fidelities of deeper circuits…
The first achievement of quantum supremacy has been claimed recently by Google for the random quantum circuit benchmark with 53 superconducting qubits. Here, we analyze the randomness of Google's quantum random-bit sampling. The heat maps of Google's random bit-strings show stripe patterns at specific qubits in contrast to the Haar-measure or classical random-bit strings. Google's data contains more bit 0 than bit 1, i.e., about 2.8\% difference, and fail to pass the NIST random number tests, while the Haar-measure or classical random-bit samples pass. Their difference is also illustrated by t…
We propose an application for near-term quantum devices: namely, generating cryptographically certified random bits, to use (for example) in proof-of-stake cryptocurrencies. Our protocol repurposes the existing "quantum supremacy" experiments, based on random circuit sampling, that Google and USTC have successfully carried out starting in 2019. We show that, whenever the outputs of these experiments pass the now-standard Linear Cross-Entropy Benchmark (LXEB), under plausible hardness assumptions they necessarily contain $Ω(n)$ min-entropy, where $n$ is the number of qubits. To achieve a net ga…
University of Edinburgh（爱丁堡大学）；CNRS（法国国家科学研究中心）；École Normale Supérieure（巴黎高等师范学院）；Sorbonne Université（索邦大学）
The search for an application of near-term quantum devices is widespread. Quantum Machine Learning is touted as a potential utilisation of such devices, particularly those which are out of the reach of the simulation capabilities of classical computers. In this work, we propose a generative Quantum Machine Learning Model, called the Ising Born Machine (IBM), which we show cannot, in the worst case, and up to suitable notions of error, be simulated efficiently by a classical device. We also show this holds for all the circuit families encountered during training. In particular, we explore quant…
In the near future, there will likely be special-purpose quantum computers with 40-50 high-quality qubits. This paper lays general theoretical foundations for how to use such devices to demonstrate "quantum supremacy": that is, a clear quantum speedup for some task, motivated by the goal of overturning the Extended Church-Turing Thesis as confidently as possible. First, we study the hardness of sampling the output distribution of a random quantum circuit, along the lines of a recent proposal by the the Quantum AI group at Google. We show that there's a natural hardness assumption, which has no…
We introduce a simple sub-universal quantum computing model, which we call the Hadamard-classical circuit with one-qubit (HC1Q) model. It consists of a classical reversible circuit sandwiched by two layers of Hadamard gates, and therefore it is in the second level of the Fourier hierarchy. We show that output probability distributions of the HC1Q model cannot be classically efficiently sampled within a multiplicative error unless the polynomial-time hierarchy collapses to the second level. The proof technique is different from those used for previous sub-universal models, such as IQP, Boson Sa…
The recent paper [F. Arute et al. Nature {\bf 574}, 505 (2019)] considered exact classical sampling of the output probability distribution of the globally depolarized random quantum circuit. In this paper, we show three results. First, we consider the case when the fidelity $F$ is constant. We show that if the distribution is classically sampled in polynomial time within a constant multiplicative error, then ${\rm BQP}\subseteq{\rm SBP}$, which means that BQP is in the second level of the polynomial-time hierarchy. We next show that for any $F\le1/2$, the distribution is classically trivially …
A crucial milestone in the field of quantum simulation and computation is to demonstrate that a quantum device can compute certain tasks that are impossible to reproduce by a classical computer with any reasonable resources. Such a demonstration is referred to as quantum supremacy. One of the most important questions is to identify setups that exhibit quantum supremacy and can be implemented with current quantum technology. The two standard candidates are boson sampling and random quantum circuits. Here, we show that quantum supremacy can be obtained in generic periodically-driven quantum many…
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 recu…
A leading proposal for verifying near-term quantum supremacy experiments on noisy random quantum circuits is linear cross-entropy benchmarking. For a quantum circuit $C$ on $n$ qubits and a sample $z \in \{0,1\}^n$, the benchmark involves computing $|\langle z|C|0^n \rangle|^2$, i.e. the probability of measuring $z$ from the output distribution of $C$ on the all zeros input. Under a strong conjecture about the classical hardness of estimating output probabilities of quantum circuits, no polynomial-time classical algorithm given $C$ can output a string $z$ such that $|\langle z|C|0^n\rangle|^2$…
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 quantu…
Noisy Intermediate-Scale Quantum (NISQ) computers are entering an era in which they can perform computational tasks beyond the capabilities of the most powerful classical computers, thereby achieving "Quantum Supremacy", a major milestone in quantum computing. NISQ Supremacy requires comparison with a state-of-the-art classical simulator. We report HPC simulations of hard random quantum circuits (RQC), which have been recently used as a benchmark for the first experimental demonstration of Quantum Supremacy, sustaining an average performance of 281 Pflop/s (true single precision) on Summit, cu…
We prove that a classical computer can efficiently sample from the photon-number probability distribution of a Gaussian state prepared by using an optical circuit that is shallow and local. Our work generalizes previous known results for qubits to the continuous-variable domain. The key to our proof is the observation that the adjacency matrices characterizing the Gaussian states generated by shallow and local circuits have small bandwidth. To exploit this structure, we devise fast algorithms to calculate loop hafnians of banded matrices. Since sampling from deep optical circuits with exponent…
Despite all the progress in quantum technologies over the last decade, there is still a dearth of practical applications for quantum computers with a small number of noisy qubits. The effort to show quantum supremacy has been largely focused on demonstrating computations that cannot be accomplished on a classical computer at all, a difficult and controversial target. Quantum advantage (a speedup over classical computers) is a more practical milestone for today's modest quantum processors. In this work, we proposed a scheme for quantum-computer compatible proof of work (cryptographic mechanism …
Diners dilemma is one of the most interesting problems in both economic and game theories. Here, we solve this problem for n (number of players) =4 with quantum rules and we are able to remove the dilemma of diners between the Pareto optimal and Nash equilibrium points of the game. We find the quantum strategy that gives maximum payoff for each diner without affecting the payoff and strategy of others. We use the quantum principles of superposition and entanglement that gives supremacy over any classical strategies. We present the circuit implementation for the game, design it on the IBM quant…
We study the problem of generating independent samples from the output distribution of Google's Sycamore quantum circuits with a target fidelity, which is believed to be beyond the reach of classical supercomputers and has been used to demonstrate quantum supremacy. We propose a new method to classically solve this problem by contracting the corresponding tensor network just once, and is massively more efficient than existing methods in obtaining a large number of uncorrelated samples with a target fidelity. For the Sycamore quantum supremacy circuit with $53$ qubits and $20$ cycles, we have g…
We describe rules to simplify quantum circuits at their boundaries, i.e. at state preparation and measurement. There, any strictly incoherent operation may be pushed into a pre- or post-processing of classical data. The rules can greatly simplify the implementation of quantum circuits and are particularly useful for hybrid algorithms on noisy intermediate-scale quantum hardware, e.g. in the context of quantum simulation. Finally we illustrate how circuit cutting can enable the rules to be applied to more locations.
Sampling from probability distributions of quantum circuits is a fundamentally and practically important task which can be used to demonstrate quantum supremacy using noisy intermediate-scale quantum devices. In the present work, we examine classical simulability of sampling from the output photon-number distribution of linear-optical circuits composed of random beam splitters with equally distributed squeezed vacuum states and single-photon states input. We provide efficient classical algorithms to simulate linear-optical random circuits and show that the algorithms' error is exponentially sm…
Classical simulations of quantum circuits are limited in both space and time when the qubit count is above 50, the realm where quantum supremacy reigns. However, recently, for the low depth circuit with more than 50 qubits, there are several methods of simulation proposed by teams at Google and IBM. Here, we present a scheme of simulation which can extract a large amount of measurement outcomes within a short time, achieving a 64-qubit simulation of a universal random circuit of depth 22 using a 128-node cluster, and 56- and 42-qubit circuits on a single PC. We also estimate that a 72-qubit ci…
The scalability, error correction and practical problem solving are important challenges for quantum computing (QC) as more emphasized by quantum supremacy (QS) experiments. Quantum path computing (QPC), recently introduced for linear optic based QCs (LOQCs) as an unconventional design, targets to obtain scalability and practical problem solving. It samples the intensity from the interference of exponentially increasing number of propagation paths obtained in multi-plane diffraction (MPD) of classical particle sources. QPC exploits MPD based quantum temporal correlations of the paths and freel…
