There is a large body of evidence for the potential of greater computational power using information carriers that are quantum mechanical over those governed by the laws of classical mechanics. But the question of the exact nature of the power contributed by quantum mechanics remains only partially answered. Furthermore, there exists doubt over the practicality of achieving a large enough quantum computation that definitively demonstrates quantum supremacy. Recently the study of computational problems that produce samples from probability distributions has added to both our understanding of the power of quantum algorithms and lowered the requirements for demonstration of fast quantum algorithms. The proposed quantum sampling problems do not require a quantum computer capable of universal operations and also permit physically realistic errors in their operation. This is an encouraging step towards an experimental demonstration of quantum algorithmic supremacy. In this paper, we will review sampling problems and the arguments that have been used to deduce when sampling problems are hard for classical computers to simulate. Two classes of quantum sampling problems that demonstrate the supremacy of quantum algorithms are BosonSampling and IQP Sampling. We will present the details of these classes and recent experimental progress towards demonstrating quantum supremacy in BosonSampling.
BosonSampling is a well-defined scheme for demonstrating quantum supremacy with photons in near term. Although relying only on multi-photon interference in nonadaptive linear-optical networks, it is hard to simulate classically. Here we study BosonSampling using matrix product states, a powerful method from quantum many-body physics. This method stores the instantaneous quantum state during the evolution of photons in the optical quantum circuit, which allows us to reveal the dynamical features of single photons in BosonSampling devices, such as entanglement entropy growth. We show the flexibl…
BosonSampling is a restricted model of quantum computation proposed recently, where a non-adaptive linear-optical network is used to solve a sampling problem that seems to be hard for classical computers. Here we show that, even if the linear-optical network has a constant number (greater than four) of beam splitter layers, the exact version of the BosonSampling problem is still classically hard, unless the polynomial hierarchy collapses to its third level. This is based on similar result known for constant-depth quantum circuits and circuits of 2-local commuting gates (IQP).
It is predicted that quantum computers will dramatically outperform their conventional counterparts. However, large-scale universal quantum computers are yet to be built. Boson sampling is a rudimentary quantum algorithm tailored to the platform of photons in linear optics, which has sparked interest as a rapid way to demonstrate this quantum supremacy. Photon statistics are governed by intractable matrix functions known as permanents, which suggests that sampling from the distribution obtained by injecting photons into a linear-optical network could be solved more quickly by a photonic experi…
Linear optics quantum computing (LOQC) is a promising approach to implementing scalable quantum computation (QC). However, this approach has very demanding physical resource requirements. Recently, Aaronson & Arkhipov showed that a simplified model, which avoids the requirement for fast feed-forward and post-selection, while likely not capable of solving BQP-complete problems efficiently, can solve an interesting sampling problem, believed to be classically hard. Loss and mode-mismatch are the dominant sources of error in such systems. We provide evidence that even lossy systems, or systems wi…
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…
We study the quantum to classical transition in Boson Sampling by analysing how $N$-boson interference is affected by inevitable noise in an experimental setup. We adopt the Gaussian noise model of Kalai and Kindler for Boson Sampling and show that it appears from some realistic experimental imperfections. We reveal a connection between noise in Boson Sampling and partial distinguishability of bosons, which allows us to prove efficient classical simulatability of noisy no-collision Boson Sampling with finite noise amplitude $ε$, i.e., $ε= Ω(1)$ as $N\to \infty$. On the other hand, using an equ…
BosonSampling is an intermediate model of quantum computation where linear-optical networks are used to solve sampling problems expected to be hard for classical computers. Since these devices are not expected to be universal for quantum computation, it remains an open question of whether any error-correction techniques can be applied to them, and thus it is important to investigate how robust the model is under natural experimental imperfections, such as losses and imperfect control of parameters. Here we investigate the complexity of BosonSampling under photon losses---more specifically, the…
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 …
BosonSampling, which we proposed three years ago, is a scheme for using linear-optical networks to solve sampling problems that appear to be intractable for a classical computer. In arXiv:1306.3995, Gogolin et al. claimed that even an ideal BosonSampling device's output would be "operationally indistinguishable" from a uniform random outcome, at least "without detailed a priori knowledge"; or at any rate, that telling the two apart might itself be a hard problem. We first answer these claims---explaining why the first is based on a definition of "a priori knowledge" so strange that, were it ad…
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 …
Quantum samplers are believed capable of sampling efficiently from distributions that are classically hard to sample from. We consider a sampler inspired by the classical Ising model. It is nonadaptive and therefore experimentally amenable. Under a plausible conjecture, classical sampling upto additive errors from this model is known to be hard. We present a trap-based verification scheme for quantum supremacy that only requires the verifier to prepare single-qubit states. The verification is done on the same model as the original sampler, a square lattice, with only a constant overhead. We ne…
Boson Sampling represents a promising approach to obtain an evidence of the supremacy of quantum systems as a resource for the solution of computational problems. The classical hardness of Boson Sampling has been related to the so called Permanent-of-Gaussians Conjecture and has been extended to some generalizations such as Scattershot Boson Sampling, approximate and lossy sampling under some reasonable constraints. However, it is still unclear how demanding these techniques are for a quantum experimental sampler. Starting from a state of the art analysis and taking account of the foreseeable …
Quantum computers are unnecessary for exponentially-efficient computation or simulation if the Extended Church-Turing thesis---a foundational tenet of computer science---is correct. The thesis would be directly contradicted by a physical device that efficiently performs a task believed to be intractable for classical computers. Such a task is BosonSampling: obtaining a distribution of n bosons scattered by some linear-optical unitary process. Here we test the central premise of BosonSampling, experimentally verifying that the amplitudes of 3-photon scattering processes are given by the permane…
Gaussian boson sampling is a promising model for demonstrating quantum computational supremacy, which eases the experimental challenge of the standard boson-sampling proposal. Here by analyzing the computational costs of classical simulation of Gaussian boson sampling,we establish a lower bound for achieving quantum computational supremacy for a class of Gaussian boson-sampling problems, where squeezed states are injected into every input mode. Specifically, we propose a method for simplifying the brute-force calculations for the transition probabilities in Gaussian boson sampling, leading to …
The recently reported experimental results claiming "quantum supremacy" achieved by Google quantum device are critically discussed. The Google team constructed a quantum chaotic system based on Josephson junction technology which cannot be reliably simulated by the present day supercomputers. However, the similar "supremacy" can be realized for properly designed micro-mechanical devices, like periodically forced Duffing oscillator, using the available technology of quartz clocks. It is also reminded that classical and quantum chaotic systems behave in a similar way. Therefore, in this case, on…
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…
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…
Boson Sampling represents a promising witness of the supremacy of quantum systems as a resource for the solution of computational problems. The classical hardness of Boson Sampling has been related to the so called Permanent-of-Gaussians Conjecture and has been extended to some generalizations such as scattershot Boson Sampling, approximate and lossy sampling under some reasonable constraints. However, it is still unclear how demanding these bounds are for a quantum experimental sampler. Starting from a state of the art analysis and focusing on the foreseeable practical conditions needed to re…
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…
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…
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…
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 …
Shortened abstract: In this thesis, I study two restricted models of quantum computing related to free identical particles. Free fermions correspond to a set of two-qubit gates known as matchgates. Matchgates are classically simulable when acting on nearest neighbors on a path, but universal for quantum computing when acting on distant qubits or when SWAP gates are available. I generalize these results in two ways. First, I show that SWAP is only one in a large family of gates that uplift matchgates to quantum universality. In fact, I show that the set of all matchgates plus any nonmatchgate p…
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 …
We propose a method to achieve quantum supremacy using ensembles of qubits, using only spin squeezing, basis rotations, and Fock state measurements. Each ensemble is assumed to be controllable only with its total spin. Using a repeated sequence of random basis rotations followed by squeezing, we show that the probability distribution of the final measurements quickly approaches a Porter-Thomas distribution. We show that the sampling probability can be related to a #P-hard problem with a complexity scaling as $(N+1)^M$, where $N$ is the number of qubits in an ensemble and $ M $ is the number of…
The Quantum Approximate Optimization Algorithm (QAOA) is designed to run on a gate model quantum computer and has shallow depth. It takes as input a combinatorial optimization problem and outputs a string that satisfies a high fraction of the maximum number of clauses that can be satisfied. For certain problems the lowest depth version of the QAOA has provable performance guarantees although there exist classical algorithms that have better guarantees. Here we argue that beyond its possible computational value the QAOA can exhibit a form of Quantum Supremacy in that, based on reasonable comple…
Computing the state of a quantum mechanical many-body system composed of indistinguishable particles distributed over a multitude of modes is one of the paradigmatic test cases of computational complexity theory: Beyond well-understood quantum statistical effects, the coherent superposition of many-particle amplitudes rapidly overburdens classical computing devices - essentially by creating extremely complicated interference patterns, which also challenge experimental resolution. With the advent of controlled many-particle interference experiments, optical set-ups that can efficiently probe ma…
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…
We study the sampling complexity of a probability distribution associated with an ensemble ofidentical noninteracting bosons undergoing a quantum random walk on a one-dimensional lattice.With uniform nearest-neighbor hopping we show that one can efficiently sample the distribution fortimes logarithmic in the size of the system, while for longer times there is no known efficient samplingalgorithm. With time-dependent hopping and optimal control, we design the time evolution toapproximate an arbitrary Haar-random unitary map analogous to that designed for photons in alinear optical network. This…
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…
BosonSampling is a quantum mechanical task involving Fock basis state preparation and detection and evolution using only linear interactions. A classical algorithm for producing samples from this quantum task cannot be efficient unless the polynomial hierarchy of complexity classes collapses, a situation believe to be highly implausible. We present method for constructing a device which uses Fock state preparations, linear interactions and Gaussian continuous-variable measurements for which one can show exact sampling would be hard for a classical algorithm in the same way as Boson Sampling. T…
We demonstrate the robustness of BosonSampling to imperfections in the linear optical network that cause a small deviation in the matrix it implements. We show that applying a noisy matrix $\tilde{U}$ that is within $ε$ of the desired matrix $U$ in operator norm leads to an output distribution that is within $εn$ of the desired distribution in variation distance, where $n$ is the number of photons. This lets us derive a sufficient tolerance each beamsplitters and phaseshifters in the network. This result considers only errors that result from the network encoding a different unitary than desir…
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…
Multiphoton interference is at the very heart of quantum foundations and applications in quantum sensing and information processing. In particular, boson sampling experiments have the potential to demonstrate quantum computational supremacy while only relying on multiphoton interference in linear optical interferometers. However, even when photonic losses are negligible, scalable experiments are challenged by the rapid decrease of the probability of success of current schemes with probabilistic sources for a large number of single photons in each experimental sample. Remarkably, we show a nove…
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…
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…
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…
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 …
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…
Fundamental questions in chemistry and physics may never be answered due to the exponential complexity of the underlying quantum phenomena. A desire to overcome this challenge has sparked a new industry of quantum technologies with the promise that engineered quantum systems can address these hard problems. A key step towards demonstrating such a system will be performing a computation beyond the capabilities of any classical computer, achieving so-called quantum supremacy. Here, using 9 superconducting qubits, we demonstrate an immediate path towards quantum supremacy. By individually tuning …
Gaussian boson sampling (GBS) allows for a way to demonstrate quantum supremacy with the relatively modest experimental resources of squeezed light sources, linear optics, and photon detection. In a realistic experimental setting, numerous effects can modify the complexity of the sampling, in particular loss, partial distinguishability of the photons, and the use of threshold detectors rather than photon counting detectors. In this paper, we investigate GBS with partial distinguishability using an approach based on virtual modes and indistinguishability efficiency. We develop a model using the…
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…
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…
Boson sampling devices are a prime candidate for exhibiting quantum supremacy, yet their application for solving problems of practical interest is less well understood. Here we show that Gaussian boson sampling (GBS) can be used for dense subgraph identification. Focusing on the NP-hard densest k-subgraph problem, we find that stochastic algorithms are enhanced through GBS, which selects dense subgraphs with high probability. These findings rely on a link between graph density and the number of perfect matchings -- enumerated by the Hafnian -- which is the relevant quantity determining samplin…
Interference of multiple photons via a linear-optical network has profound applications for quantum foundation, quantum metrology and quantum computation. Particularly, a boson sampling experiment with a moderate number of photons becomes intractable even for the most powerful classical computers, and will lead to "quantum supremacy". Scaling up from small-scale experiments requires highly indistinguishable single photons, which may be prohibited for many physical systems. Here we experimentally demonstrate a time-resolved version of boson sampling by using photons not overlapping in their fre…
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…
The demonstration of quantum speedup, also known as quantum computational supremacy, that is the ability of quantum computers to outperform dramatically their classical counterparts, is an important milestone in the field of quantum computing. While quantum speedup experiments are gradually escaping the regime of classical simulation, they still lack efficient verification protocols and rely on partial validation. Here we derive an efficient protocol for verifying with single-mode Gaussian measurements the output states of a large class of continuous-variable quantum circuits demonstrating qua…
Boson Sampling is the problem of sampling from the same output probability distribution as a collection of indistinguishable single photons input into a linear interferometer. It has been shown that, subject to certain computational complexity conjectures, in general the problem is difficult to solve classically, motivating optical experiments aimed at demonstrating quantum computational "supremacy". There are a number of challenges faced by such experiments, including the generation of indistinguishable single photons. We provide a quantum circuit that simulates bosonic sampling with arbitrar…
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…
