Quantum matchgate computations and linear threshold gates

Abstract

The theory of matchgates is of interest in various areas in physics and computer science. Matchgates occur, for example, in the study of fermions and spin chains, in the theory of holographic algorithms and in several recent works in quantum computation. In this paper, we completely characterize the class of Boolean functions computable by unitary two-qubit matchgate circuits with some probability of success. We show that this class precisely coincides with that of the linear threshold gates. The latter is a fundamental family that appears in several fields, such as the study of neural networks. Using the above characterization, we further show that the power of matchgate circuits is surprisingly trivial in those cases where the computation is to succeed with high probability. In particular, the only functions that are matchgate-computable with success probability greater than 3/4 are functions depending on only a single bit of the input.

Compressed simulation of evolutions of theXYmodel

“…Strong simulation means that the probabilities of the measurement outcomes is computed efficiently exactly, whereas weak simulation means that one can sample from this probability distribution classically efficiently. Those two notions are fundamentally different, and quantum computations which cannot be simulated strongly might well be weakly simulatable [36]. Note that in the compressed simulation considered here, the probabilities of the measurement outcomes of both the circuits of width n and the one of width log(n) coincide.…”

“…The reason why this compressed way of quantum simulation works is because all the circuits investigated here, were match-gate circuits, for which it has been shown that their power coincides with a universal quantum computer of exponentially smaller width. It should be noted that any computation which can be simulated in the strong sense by an exponentially smaller system, as done here, must be classically efficiently simulatable since the dimension of the Hilbert space describing the system is linear in n. Regarding classical simulation one distinguishes between a strong simulation and a weak simulation [36]. Strong simulation means that the probabilities of the measurement outcomes is computed efficiently exactly, whereas weak simulation means that one can sample from this probability distribution classically efficiently.…”

Solving search problems by strongly simulating quantum circuits

“…It turns out that such languages are trivial; whether or not a word is part of the language can be decided by considering at most a single bit of the word33.…”

“…Our formalism also allows the efficient finding of words in languages decided by a Gaussian circuit followed by a computational basis measurement in a single qubit. It turns out that such languages are trivial; whether or not a word is part of the language can be decided by considering at most a single bit of the word [33].…”

Jordan-Wigner formalism for arbitrary 2-input 2-output matchgates and their classical simulation

“…Furthermore, for matchgate circuits we consider only unitary circuits, without intermediate measurements, having measurements only at the end to provide output probabilities. The computational power of such unitary matchgate circuits has been shown in [16] to coincide with that of space-bounded quantum computation, and further results on the ability of these circuits to compute Boolean functions have been given in [17]. Some classical simulation results for adaptive matchgate circuits have been given in [3].…”