In his famous 1982 lecture, Nobel laureate Richard Feynman seeded the idea of quantum computation. He reasoned that the amount of classical information required to describe a quantum system scaled exponentially with its size, and thus could not be efficiently simulated by a classical computer. Indeed, a fairly modest system of 100 quantum bits would already require more information to describe than all the present data storage on Earth [1]. Given this, there would likely be computational tasks that achieve quantum supremacy—impossible to perform practically without the aid of a quantum information processor. In the 37 intervening years, these initial ideas have solidified, as the discovery of a string of quantum algorithms—from efficient factoring of large numbers to simulating chemical reactions—fueled a race for quantum supremacy. Yet, even theoretically, conclusively deciding—if quantum supremacy has been achieved—is non-trivial. Judging what problems can be efficiently solved by classical computers turns out to be exceedingly hard in itself. Take factoring, for instance. It is really classical intractable, or is it merely that we have not yet found an efficient classical means to factor? As we better understood how quantum computers process information, we also got better at finding classical methods to simulate them. This in turn has inspired new efficient classical algorithms in surprising situations [2]. Prof. Man-Hong Yung, associate professor at the Southern University of Science and Technology, and colleagues provided a beautiful illustration of such a discovery. Their work has been published in the July 2019 issue of National Science Review [3]. It considers a specialized type of linear optical quantum processor. The input and output are encoded within an array of optical modes by adjusting their initial photon number. Computation is achieved by interfering with these modes via a network of beam splitters, such that photons in one mode are transferred to another. Such processors have captured scientific attention due to their ability to perform a task known as boson sampling. When each mode is initialized with a single photon, the output statistics allows efficient evaluation of the matrix permanent—a computational problem with very strong evidence for being classically intractable [4]. This, coupled with their relative ease of engineering, made this specialized type of quantum processor an ideal holy grail for demonstrating quantum supremacy. Given this, one may expect that classical computers can say little about what such processors will output. After all, the number of paths that a photon can take to reach one mode from another scales exponentially with the size of the beam-splitter network. One may therefore assume that it would take exponential resources to track these paths to determine the probability of a photon moving from mode A to mode B. However, Yung and colleagues showed that a classical algorithm can efficiently estimate the transition probabilities—as long as one allows for additive error. This result represents a significant generalization of prior results by Scott Aaronson, which considers the case where there exists at most one photon in each mode [5]. In contrast, Yung's solution works regardless of the number of photons encoded in each mode. The result not only benefits the simulation of bosonic interference, but also has strong consequences in computing. Suppose that we use such specialized quantum processors to make binary decisions. That is, depending on input, we feed in particular distributions of photons—and then decide to take a particular action depending on whether the probability of seeing x photons in a particular output is above some threshold. In such contexts, a classical algorithm that estimates this probability up to the bounded additive error is sufficient for making identical decisions as the quantum processer itself. As such, Yung's algorithm enables a classical computer to be able to efficiently mimic the decisions of its specialized linear optical counterpart. This will no doubt have important ramifications in the present race for quantum supremacy. Much present effort is spent on demonstrating quantum supremacy through boson sampling, and the results here indicate that we will need to introduce more sophisticated quantum architectures if we wish to demonstrate such supremacy in making the right decisions.
In his famous 1982 lecture, Nobel laureate Richard Feynman seeded the idea of quantum computation. He reasoned that the amount of classical information required to describe a quantum system scaled exponentially with its size, and thus could not be efficiently simulated by a classical computer. Indeed, a fairly modest system of 100 quantum bits would already require more information to describe than all the present data storage on Earth [1]. Given this, there would likely be computational tasks that achieve quantum supremacy—impossible to perform practically without the aid of a quantum information processor. In the 37 intervening years, these initial ideas have solidified, as the discovery of a string of quantum algorithms—from efficient factoring of large numbers to simulating chemical reactions—fueled a race for quantum supremacy. Yet, even theoretically, conclusively deciding—if quantum supremacy has been achieved—is non-trivial. Judging what problems can be efficiently solved by classical computers turns out to be exceedingly hard in itself. Take factoring, for instance. It is really classical intractable, or is it merely that we have not yet found an efficient classical means to factor? As we better understood how quantum computers process information, we also got better at finding classical methods to simulate them. This in turn has inspired new efficient classical algorithms in surprising situations [2]. Prof. Man-Hong Yung, associate professor at the Southern University of Science and Technology, and colleagues provided a beautiful illustration of such a discovery. Their work has been published in the July 2019 issue of National Science Review [3]. It considers a specialized type of linear optical quantum processor. The input and output are encoded within an array of optical modes by adjusting their initial photon number. Computation is achieved by interfering with these modes via a network of beam splitters, such that photons in one mode are transferred to another. Such processors have captured scientific attention due to their ability to perform a task known as boson sampling. When each mode is initialized with a single photon, the output statistics allows efficient evaluation of the matrix permanent—a computational problem with very strong evidence for being classically intractable [4]. This, coupled with their relative ease of engineering, made this specialized type of quantum processor an ideal holy grail for demonstrating quantum supremacy. Given this, one may expect that classical computers can say little about what such processors will output. After all, the number of paths that a photon can take to reach one mode from another scales exponentially with the size of the beam-splitter network. One may therefore assume that it would take exponential resources to track these paths to determine the probability of a photon moving from mode A to mode B. However, Yung and colleagues showed that a classical algorithm can efficiently estimate the transition probabilities—as long as one allows for additive error. This result represents a significant generalization of prior results by Scott Aaronson, which considers the case where there exists at most one photon in each mode [5]. In contrast, Yung's solution works regardless of the number of photons encoded in each mode. The result not only benefits the simulation of bosonic interference, but also has strong consequences in computing. Suppose that we use such specialized quantum processors to make binary decisions. That is, depending on input, we feed in particular distributions of photons—and then decide to take a particular action depending on whether the probability of seeing x photons in a particular output is above some threshold. In such contexts, a classical algorithm that estimates this probability up to the bounded additive error is sufficient for making identical decisions as the quantum processer itself. As such, Yung's algorithm enables a classical computer to be able to efficiently mimic the decisions of its specialized linear optical counterpart. This will no doubt have important ramifications in the present race for quantum supremacy. Much present effort is spent on demonstrating quantum supremacy through boson sampling, and the results here indicate that we will need to introduce more sophisticated quantum architectures if we wish to demonstrate such supremacy in making the right decisions.
The strong efforts towards the realization of scalable quantum computing are motivated by the promise of super-polynomial speed-up in several computational tasks. A first fundamental step in this route is to provide concrete evidence of ‘quantum supremacy'. Such a regime is obtained when a quantum device is capable of solving a specific problem faster than the best classical hardware. A serious attempt to provide a first demonstration of quantum speed-up with affordable resources is represented by boson sampling [1,2], a non-universal quantum-computational protocol exploiting indistinguishable single photons and linear optical devices. In this context, strong theoretical evidence has been reported that the interference pattern generated by a boson sampler is hard to simulate with classical resources, this pattern being related to the calculation of a hard-to-compute quantity known as a matrix permanent. Formal achievement of the regime of quantum supremacy will answer two very relevant questions: Are we able to benefit from the power of quantum mechanics or does there exist a fundamental reason why this achievement is forbidden? Once this regime has been achieved, are we able to certify the correct functioning of the quantum boson samplers or do we face an undecidability problem? Starting from these intriguing and promising theoretical results, an experimental race kicked off with the aim of reaching the quantum supremacy regime with boson sampling by employing a photonic platform [3]. A large amount of effort has been devoted to pushing the limits of current photonic systems [1], in order to significantly increase the size (number of photons and optical modes) that can be handled with a quantum device. In parallel to this quest for improved quantum performance, it is fundamental to benchmark the limits of a classical simulation of boson sampling. More specifically, in the race for quantum supremacy, what is the best possible classical simulation that can be performed? Such an analysis would be necessary to define the computational capacity possessed by such a classical agent to perform the task (see Fig. 1). In the race for quantum supremacy, a user who employs the best classical hardware and software, and a second agent who exploits a quantum device compete to solve the boson sampling task faster. In a recent article, Wu et al. [4] have provided a fundamental step to define and push the limits of a classical simulation of boson sampling. The authors have employed Tianhe-2, one of the world's most powerful super-computers, to perform such a task. More specifically, by using such high-performance computing hardware and by employing the fastest algorithm to date, Wu et al. have shown that calculating the permanent of a |50×50| matrix can be performed in 100 min. Recent results [5] have shown that this computation corresponds to the classical hardness of drawing a 50-photon sample from a boson sampling distribution. The strong computational capacity of Tianhe-2 also allowed us to address the precision that the known fastest classical algorithms can reach in such a large-scale calculation. This latter aspect is indeed of fundamental importance, since any reliable classical computation must be accompanied by an appropriate error analysis. The study reported in Ref. [4] represents by far the largest calculation to date related to the classical simulation of boson sampling, whose relevance can be found in two different but related aspects. First, it raises the bar for any quantum implementation aimed at reaching quantum advantage, and provides a fundamental benchmark to which any photonic hardware must compare. This result suggests that a larger effort must be dedicated to pushing quantum technologies in this direction. Second, improved classical simulation also has important consequences in the context of boson sampling certification, i.e. the verification that a quantum device is correctly performing the required task. Indeed, first validation procedures in the quantum advantage regime must be assisted by powerful classical computational resources that, while requiring longer processing time than quantum hardware, will necessarily have to perform complex calculations.
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!
We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P^#P=BPP^NP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation. Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the "Permanent-of-Gaussians Conjecture", which says that it is #P-hard to approximate the permanent of a matrix A of independent N(0,1) Gaussian entries, with high probability over A; and the "Permanent Anti-Concentration Conjecture", which says that |Per(A)|>=sqrt(n!)/poly(n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application. This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists.
It remains questionable whether all physically realizable computational model can be simulated by currently available digital computer in an efficient manner. The theory of Quantum Mechanics offers an answer to this question through one of its branches that intersects with Computer Science, known as Quantum Computing. It claims that current computers, which is operating based on classical mechanics cannot efficiently simulate quantum mechanical process. Under this claim, a computer that is operating based on quantum mechanics holds more computational power than classical computers, promising a possibility to computationally solve problems that are classically intractable. To show that this claim holds, numerous quantum computational models have been proposed and analyzed through the lens of Computational Complexity Theory, providing a rigorous notion of efficiency. Models that are more feasible experimentally, although not necessarily universal, have also been of a particular interests given the numerous complicated challenges surrounding the experimental implementation of a universal quantum computer. One of this non-universal models is called Gaussian Boson Sampling (GBS), which embraces the methods of Quantum Optics by sampling the number of photon outputs from a linear optical network as a means to process quantum information. Compared to its prototype model, Boson Sampling, GBS is more experimentally feasible due to its method of generating photon that is based on the Continuous Variable (CV) paradigm of Quantum Information. Nevertheless as with Boson Sampling, it was also shown that classical simulation of GBS cannot be performed efficiently under some highly plausible conjectures. In the work presented in this thesis, we probe the complexity of classically constructing the probability distribution induced by GBS using the method of Quantum Homodyne Tomography. Instead of photon number measurement, this method further embraces the CV paradigm in the measurement of GBS by using Homodyne measurements that outputs real numbers. Sample data obtained from the measurement outcomes is then classically processed using Pattern Functions, which we use to approximate the probability distribution of GBS. A series of computer simulations that approximates some probability values of GBS are presented here, which shows that the number of samples needed to decrease the error of the approximation increases rapidly for more accurate approximation. We have derived a bound for the approximation error which can be achieved given that the sampled data satisfies a certain statistical property. This statistical property can eventually be satisfied as the number of Homodyne samples increases. Having shown another evidence for the complexity of approximating GBS using Homodyne Tomography, we further investigate the nature of the complexity of GBS by analyzing its classical-quantum computational boundary in the spectrum of amount of noise present in a GBS setup. In doing this, we built our discussions on the measure of non-classicality using the presence of negative values in the Quasi-probability distribution (QPD) representation of states of a quantum system. We then discuss existing results that shows the possibility of efficient classical simulation of a quantum computational model given a sufficient amount of noise. These results impose a necessary criteria of the minimum amount of noise that an experiment has to surpass in order for it to be impossible to simulate classically in an efficient manner. We extend this idea for GBS, where we propose a modified GBS where we consider noise in its measurements that opens a way for a possible formulation of a noise condition that a GBS experiment has to fulfill to rule out possible efficient classical simulation. To further probe the classical-quantum computational boundary, we also propose a different quantum computational model called the Photon-added Thermal State CV-Sampling (PATS CVS), which can be formulated similarly to our noisy GBS model. We then discuss existing results that analyzes the negativity of the QPD representation of a noisy Photon-added Thermal state (PATS), which constitutes the inputs for PATS CVS. Using the result that shows the quantum-classical transition in terms of the QPD negativity for a PATS state when the amount of noise changes, we discuss the potential of PATS CVS model as a possible avenue to further probe the transition between efficiently simulable and non-efficiently simulable quantum mechanical experiment.
We study the classical complexity of the exact Boson Sampling problem where the objective is to produce provably correct random samples from a particular quantum mechanical distribution. The computational framework was proposed in STOC ’11 by Aaronson and Arkhipov in 2011 as an attainable demonstration of ‘quantum supremacy’, that is a practical quantum computing experiment able to produce output at a speed beyond the reach of classical (that is non-quantum) computer hardware. Since its introduction Boson Sampling has been the subject of intense international research in the world of quantum computing. On the face of it, the problem is challenging for classical computation. Aaronson and Arkhipov show that exact Boson Sampling is not efficiently solvable by a classical computer unless P#P = BPPNP and the polynomial hierarchy collapses to the third level. The fastest known exact classical algorithm for the standard Boson Sampling problem requires time to produce samples for a system with input size n and m output modes, making it infeasible for anything but the smallest values of n and m. We give an algorithm that is much faster, running in time and additional space. The algorithm is simple to implement and has low constant factor overheads. As a consequence our classical algorithm is able to solve the exact Boson Sampling problem for system sizes far beyond current photonic quantum computing experimentation, thereby significantly reducing the likelihood of achieving near-term quantum supremacy in the context of Boson Sampling.
We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P#P=BPPNP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation.