Simulating quantum circuits using efficient tensor network contraction algorithms with subexponential upper bound

We derive a rigorous upper bound on the classical computation time of finite-ranged tensor network contractions in $d \geq 2$ dimensions. By means of the Sphere Separator Theorem, we are able to take advantage of the structure of quantum circuits to speed up contractions to show that quantum circuits of single-qubit and finite-ranged two-qubit gates can be classically simulated in subexponential time in the number of gates. In many practically relevant cases this beats standard simulation schemes. Moreover, our algorithm leads to speedups of several orders of magnitude over naive contraction schemes for two-dimensional quantum circuits on as little as an $8 \times 8$ lattice. We obtain similarly efficient contraction schemes for Google's Sycamore-type quantum circuits, instantaneous quantum polynomial-time circuits and non-homogeneous (2+1)-dimensional random quantum circuits.Comment: 8 pages, 6 figure

The power of restricted quantum computational models

Restricted models of quantum computation are ones that have less power than a universal quantum computer. We studied the consequences of removing particular properties from a universal quantum computer to discover whether those resources were important.In the first part of the thesis we studied universal quantum computers which are implemented using Clifford gates, adaptive measurements, and magic states. The Gottesman–Knill theorem shows that circuits in this form which do not use magic states can be simulated by a classical computer. We extended this result to show that all circuits in this form can be partially simulated; the same computation can be implemented using a smaller quantum computer with the assistance of some polynomial time classical computation. We also identified a subclass of these computations that can be shown to not be entirely classically simulated by any method, given certain complexity theoretic assumptions are true.In the next part of the thesis we examine the role of entanglement in noisy quantum computations. Entanglement is necessary for noiseless quantum computers to have any quantum advantage, but it is not known whether the same is true for mixed state quantum computers. We show that entanglement, unexpectedly, does play a crucial role in the most well known mixed state computer: the one clean qubit model.Finally, we investigate how closely classical simulation is related to another idea of classicality.This notion captures how easily the final state of a computation can be learnt, given samples of measurements from it. We find an extra condition under which a circuit that is classically simulable is also efficiently learnable

Matchgates and classical simulation of quantum circuits

Let G(A,B) denote the 2-qubit gate which acts as the 1-qubit SU(2) gates A and B in the even and odd parity subspaces respectively, of two qubits. Using a Clifford algebra formalism we show that arbitrary uniform families of circuits of these gates, restricted to act only on nearest neighbour (n.n.) qubit lines, can be classically efficiently simulated. This reproduces a result originally proved by Valiant using his matchgate formalism, and subsequently related by others to free fermionic physics. We further show that if the n.n. condition is slightly relaxed, to allowing the same gates to act only on n.n. and next-n.n. qubit lines, then the resulting circuits can efficiently perform universal quantum computation. From this point of view, the gap between efficient classical and quantum computational power is bridged by a very modest use of a seemingly innocuous resource (qubit swapping). We also extend the simulation result above in various ways. In particular, by exploiting properties of Clifford operations in conjunction with the Jordan-Wigner representation of a Clifford algebra, we show how one may generalise the simulation result above to provide further classes of classically efficiently simulatable quantum circuits, which we call Gaussian quantum circuits.Comment: 18 pages, 2 figure

Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy

We consider quantum computations comprising only commuting gates, known as IQP computations, and provide compelling evidence that the task of sampling their output probability distributions is unlikely to be achievable by any efficient classical means. More specifically we introduce the class post-IQP of languages decided with bounded error by uniform families of IQP circuits with post-selection, and prove first that post-IQP equals the classical class PP. Using this result we show that if the output distributions of uniform IQP circuit families could be classically efficiently sampled, even up to 41% multiplicative error in the probabilities, then the infinite tower of classical complexity classes known as the polynomial hierarchy, would collapse to its third level. We mention some further results on the classical simulation properties of IQP circuit families, in particular showing that if the output distribution results from measurements on only O(log n) lines then it may in fact be classically efficiently sampled.Comment: 13 page

Classical simulations of Abelian-group normalizer circuits with intermediate measurements

Quantum normalizer circuits were recently introduced as generalizations of Clifford circuits [arXiv:1201.4867]: a normalizer circuit over a finite Abelian group $G$ is composed of the quantum Fourier transform (QFT) over G, together with gates which compute quadratic functions and automorphisms. In [arXiv:1201.4867] it was shown that every normalizer circuit can be simulated efficiently classically. This result provides a nontrivial example of a family of quantum circuits that cannot yield exponential speed-ups in spite of usage of the QFT, the latter being a central quantum algorithmic primitive. Here we extend the aforementioned result in several ways. Most importantly, we show that normalizer circuits supplemented with intermediate measurements can also be simulated efficiently classically, even when the computation proceeds adaptively. This yields a generalization of the Gottesman-Knill theorem (valid for n-qubit Clifford operations [quant-ph/9705052, quant-ph/9807006] to quantum circuits described by arbitrary finite Abelian groups. Moreover, our simulations are twofold: we present efficient classical algorithms to sample the measurement probability distribution of any adaptive-normalizer computation, as well as to compute the amplitudes of the state vector in every step of it. Finally we develop a generalization of the stabilizer formalism [quant-ph/9705052, quant-ph/9807006] relative to arbitrary finite Abelian groups: for example we characterize how to update stabilizers under generalized Pauli measurements and provide a normal form of the amplitudes of generalized stabilizer states using quadratic functions and subgroup cosets.Comment: 26 pages+appendices. Title has changed in this second version. To appear in Quantum Information and Computation, Vol.14 No.3&4, 201

Quantum Commuting Circuits and Complexity of Ising Partition Functions

Instantaneous quantum polynomial-time (IQP) computation is a class of quantum computation consisting only of commuting two-qubit gates and is not universal in the sense of standard quantum computation. Nevertheless, it has been shown that if there is a classical algorithm that can simulate IQP efficiently, the polynomial hierarchy (PH) collapses at the third level, which is highly implausible. However, the origin of the classical intractability is still less understood. Here we establish a relationship between IQP and computational complexity of the partition functions of Ising models. We apply the established relationship in two opposite directions. One direction is to find subclasses of IQP that are classically efficiently simulatable in the strong sense, by using exact solvability of certain types of Ising models. Another direction is applying quantum computational complexity of IQP to investigate (im)possibility of efficient classical approximations of Ising models with imaginary coupling constants. Specifically, we show that there is no fully polynomial randomized approximation scheme (FPRAS) for Ising models with almost all imaginary coupling constants even on a planar graph of a bounded degree, unless the PH collapses at the third level. Furthermore, we also show a multiplicative approximation of such a class of Ising partition functions is at least as hard as a multiplicative approximation for the output distribution of an arbitrary quantum circuit.Comment: 36 pages, 5 figure