Average-Case Complexity Versus Approximate Simulation of Commuting Quantum Computations

Abstract

We use the class of commuting quantum computations known as IQP (Instantaneous Quantum Polynomial time) to strengthen the conjecture that quantum computers are hard to simulate classically. We show that, if either of two plausible average-case hardness conjectures holds, then IQP computations are hard to simulate classically up to constant additive error. One conjecture relates to the hardness of estimating the complextemperature partition function for random instances of the Ising model; the other concerns approximating the number of zeroes of random low-degree polynomials. We observe that both conjectures can be shown to be valid in the setting of worst-case complexity. We arrive at these conjectures by deriving spin-based generalisations of the Boson Sampling problem that avoid the so-called permanent anticoncentration conjecture.Quantum computers are conjectured to outperform classical computers for a variety of important tasks ranging from integer factorisation [1] to the simulation of quantum mechanics [2]. However, to date there is relatively little rigorous evidence for this conjecture. It is well established that quantum computers can yield an exponential advantage in the query and communication complexity models. But in the more physically meaningful model of time complexity, there are no proven separations known between quantum and classical computation.This can be seen as a consequence of the extreme difficulty of proving bounds on the power of classical computing models, such as the famous P vs. NP problem. Given this difficulty, the most we can reasonably hope for is to show that quantum computations cannot be simulated efficiently classically, assuming some widely believed complexity-theoretic conjecture. For example, any set of quantum circuits that can implement Shor's algorithm [1] provides a canonical example, with the unlikely consequence of efficient classical simulation of this class of quantum circuits being the existence of an efficient classical factoring algorithm. However, one could hope for the existence of other examples that have wider-reaching complexity-theoretic consequences.With this in mind, in both [3] and [4] it was shown that the existence of an efficient classical sampler from a distribution that is close to the output distribution of an arbitrary quantum circuit, to within a small multiplicative error in each output probability, would imply that post-selected classical computation is equivalent to post-selected quantum computation. This consequence is considered very unlikely as it would collapse the infinite tower of complexity classes known as the Polynomial Hierarchy [5] to its third level. In both works this was proven even for non-universal quantum circuit families: commuting quantum circuits in the case of [3], and linear-optical networks in [4]. These non-universal families are of physical interest because they are simpler to implement, and easier to analyse because of the elegant mathematical structures on which they are based. * michael.bremner@uts.edu.au Unfortu...

Ancilla-driven instantaneous quantum polynomial time circuit for quantum supremacy

“…In 2013, Aaronson and Arkhipov showed this for boson sampling under two conjectures [11]. By generalizing their argument, Bremner et al showed that IQP circuits are unlikely to be simulated classically up to an error in the l 1 norm whose value is constant under only one conjecture [6].…”

“…Unlike the argument in Ref. [6], in the argument to prove theorem 6, the initial state of each of the ancillary qubits is set to |0 to construct C (2) f . In other words, if we construct C (2) f without setting the initial state of each of them to |0 when f is chosen uniformly at random, we can prove theorem 6 for ǫ =const.…”

“…[6], for any degree-3 polynomial f : {0, 1} n → {0, 1} over F 2 satisfying f (0 n ) = 0, there is an IQP circuit C f composed of the controlled-controlled-Z gate Λ(Λ(Z)), Λ(Z), and Z, and it satisfies 0| ⊗n C f |0 ⊗n = gap(f )/2 n .…”

“…To do so, we use an argument similar to that used in Ref. [6]. In the following, BPP NP is a complexity class consisting of decision problems that can be solved by randomized classical polynomial-time computation given an oracle that can solve any NP problem.…”

“…Recently, in order to develop an understanding of the difference between them, the classical simulatability of restricted quantum computation has been extensively studied. For such quantum computation, commuting quantum computation including instantaneous quantum polynomial time (IQP) [2][3][4][5][6][7], deterministic quantum computation with 1 pure qubit (DQC1) [8][9][10], boson sampling [11][12][13][14], constant-depth quantum circuit [15,16], and permutational quantum computing [17] have been proposed.…”