Abstract We define rewinding operators that invert quantum measurements. Then, we define complexity classes $$\textsf{RwBQP}$$ RwBQP , $$\textsf{CBQP}$$ CBQP , and $$\textsf{AdPostBQP}$$ AdPostBQP as sets of decision problems solvable by polynomial-size quantum circuits with a polynomial number of rewinding operators, cloning operators, and adaptive postselections, respectively. Our main result is that $$\textsf{BPP}^\textsf{PP}\subseteq \textsf{RwBQP}=\textsf{CBQP}=\textsf{AdPostBQP}\subseteq \textsf{PSPACE}$$ BPP PP ⊆ RwBQP = CBQP = AdPostBQP ⊆ PSPACE . As a byproduct of this result, we show that any problem in $$\textsf{PostBQP}$$ PostBQP can be solved with only postselections of events that occur with probabilities polynomially close to one. Under the strongly believed assumption that $$\textsf{BQP}\nsupseteq \textsf{SZK}$$ BQP ⊉ SZK , or the shortest independent vectors problem cannot be efficiently solved with quantum computers, we also show that a single rewinding operator is sufficient to achieve tasks that are intractable for quantum computation. Finally, we show that rewindable Clifford circuits remain classically simulatable, but rewindable instantaneous quantum polynomial time circuits can solve any problem in $$\textsf{PP}$$ PP .
Abstract We define rewinding operators that invert quantum measurements. Then, we define complexity classes RwBQP RwBQP , CBQP CBQP , and AdPostBQP AdPostBQP as sets of decision problems solvable by polynomial-size quantum circuits with a polynomial number of rewinding operators, cloning operators, and adaptive postselections, respectively. Our main result is that BPPPP⊆RwBQP=CBQP=AdPostBQP⊆PSPACE BPP PP ⊆ RwBQP = CBQP = AdPostBQP ⊆ PSPACE . As a byproduct of this result, we show that any problem in PostBQP PostBQP can be solved with only postselections of events that occur with probabilities polynomially close to one. Under the strongly believed assumption that BQP⊉SZK BQP ⊉ SZK , or the shortest independent vectors problem cannot be efficiently solved with quantum computers, we also show that a single rewinding operator is sufficient to achieve tasks that are intractable for quantum computation. Finally, we show that rewindable Clifford circuits remain classically simulatable, but rewindable instantaneous quantum polynomial time circuits can solve any problem in PP PP .
We consider restricted versions of postBQP where postselection probabilities can be efficiently calculated by classical or quantum computers. We show that such restricted postBQP classes are in AWPP(⊆PP=postBQP). This result suggests that postselecting an event with an exponentially small probability does not necessarily boost BQP to PP. The best upperbound of BQP is AWPP, and therefore restricted postBQP classes are other examples of complexity classes slightly above BQP. In [C. M. Lee and J. Barrett, arXiv:1412.8671], it was shown that the computational capacity of a general probabilistic theory which satisfies the tomographic locality is in AWPP. Our result therefore implies that quantum physics with restricted postselections is an example of a super quantum theory which seems to be outside of their general probabilistic theory but its computational capacity is also in AWPP. Although it is physically natural to expect that such restricted postBQP classes are not equivalent to BQP, we show that UP∩coUP, which is unlikely to be in BQP, is contained in the restricted postBQP classes. Finally, we also consider another restricted class of postBQP, where the postselected probability depends only on the size of inputs. We show that this restricted version is contained in APP, where AWPP⊆APP⊆PP=postBQP.
We demonstrate the superior capabilities of the recently proposed Lorentz quantum computer (LQC) compared to conventional quantum computers. We introduce an associated computational complexity class termed bounded-error Lorentz quantum polynomial-time (BLQP), demonstrating its equivalence to the complexity class P♯P. We present LQC algorithms that efficiently solve the problem of maximum independent set, PP (probabilistic polynomial-time), and consequently P♯P, all within polynomial time. Additionally, we show that the quantum computing with postselection proposed by Aaronson can be efficiently simulated by LQC, but not vice versa.
Depending on time, I'll discuss some subset of the following results. (1) There exists an oracle relative to which classical and quantum computers can solve the same approximate sampling problems in polynomial time, and yet the polynomial hierarchy is infinite. This means that any "BosonSampling-type theorem" will need to be non-relativizing. (2) Can P be separated from BQP by an efficiently-computable oracle (one in P/poly)? Yes if quantum-resistant one-way functions exist; no if P=PSPACE. The latter builds on recent results in quantum query complexity by myself and Shalev Ben-David. (3) How hard is it to approximately sample the output distribution of a random quantum circuit? One can do it (and even calculate the probabilities) in polynomial space and m^O(n) time, where m is the number of gates and n is the number of qubits. But if that's essentially optimal for estimating probabilities, then approximate sampling also requires exponential time. Joint work with Lijie Chen (Tsinghua University)
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 explore the space "just above" BQP by defining a complexity class naCQP (non-adaptive Collapse-free Quantum Polynomial time) which is larger than BQP but does not contain NP relative to an oracle. The class is defined by imagining that quantum computers can perform (non-adaptive) measurements that do not collapse the wavefunction. This non-physical model of computation can efficiently solve problems such as Graph Isomorphism and Approximate Shortest Vector which are believed to be intractable for quantum computers. Furthermore, it can search an unstructured N-element list in Õ(N1/3) time, but no faster than Ω(N1/4), and hence cannot solve NP-hard problems in a black box manner. In short, this model of computation is more powerful than standard quantum computation, but only slightly so. This is surprising as most modifications of BQP increase the power of quantum computation to NP or beyond.
Non-Hermitian (NH) quantum systems demonstrate striking differences from their Hermitian counterparts, leading to claims of NH advantage in areas ranging from metrology to entanglement generation. We show that in the context of quantum computation, any such NH advantage is unlikely to be scalable as an efficient computational resource: If coherent nonunitary evolution with renormalization could be realized with only polynomial overhead, then the resulting model could implement postselection, implying implausibly strong complexity-theoretic power under standard assumptions. We define NHBQP ( U ) as the computational power of polynomial-size quantum circuits that, in addition to a standard universal unitary gate set, may apply a fixed gate U on O ( 1 ) qubits that is not proportional to a unitary, with the state renormalized after each use of U . We prove that this model is powerful enough to decide every language in PostBQP (equivalently PP ). Moreover, in the standard uniform circuit-family model, this characterization is tight: For any fixed such nonunitary gate U , NHBQP ( U ) = PostBQP = PP . PostBQP is believed intractable, so this suggests that any scalable NH computational advantage must come with a compensating cost limiting its efficiency. Additionally, we study simulation complexity of restricted classes of nonunitary systems by purifying them to postselected unitary evolution in a form preserving system locality. Using this framework, we show that unitary gates with postselection can simulate not only evolution under NH Hamiltonians but arbitrary quantum trajectories. Any NH model whose purification lies in a strongly simulable unitary family (e.g., Clifford, matchgate, or low-bond-dimension tensor-network circuits) remains efficiently classically simulable, provided the relevant postselected events occur with probability Ω ( 2 − poly ( n ) ) . Thus, adding non-Hermiticity to a universal unitary system makes it infeasibly computationally powerful, while adding it to a strongly simulable system adds no computational power in this setting.