Quantum matchgate computations and linear threshold gates

Of particular interest, also in recent work, is the class of quantum processes generated by matchgates [5,8,9,10,11,12,13]. The latter are a class of unitary two-qubit operations that are defined by certain algebraic constraints. The theory of matchgates is an instance of a research area that displays strong connections to both physics and computer science [5,8,9,10,11,12,13]. In the study of strongly correlated systems, for example, the dynamics of an important class of 1D quantum systems such as the XY model are modeled by matchgate circuits i.e. for such hamiltonians H one can construct a poly-size matchgate circuit C t such that C t = e itH for any time t (see e.g. [10]). Employing mappings between spin- 12 systems and fermions, matchgate circuits further describe the dynamics of all non-interacting fermionic systems [8]. In the theory of quantum computation, matchgates are of particular interest as they provide a key example of class of nontrivial quantum circuits that cannot offer any speed-up over classical computers (in spite of e.g. the complex entangled states such circuits may generate) [5,10]. In addition, matchgate computations were recently found to be equivalent to space-bounded universal quantum computation [12]. In classical computer science, finally, matchgates occur in various studies related to e.g. the theory of holographic algorithms [5,9].

The aim of the present paper is to characterize the computational power of matchgate circuits. We will in particular study which boolean functions 1 can be computed with such circuits. Given an arbitrary matchgate circuit U , the question is asked which boolean function f (x) can be computed (probabilistically) by initializing the system in the computational basis state |x, 0 (where x represents an input string and 0 is a string of ancillary zeros), by subsequently running the circuit U and finally measuring, say, the first qubit. This setting captures in perhaps the most elementary way the computational power of matchgate circuits, associating which each circuit a yes/no question as commonly done.

We remark that, beyond its natural computer scientific interest, such an investigation is relevant from a intrinsic physical perspective as well. In particular, given the aforementioned equivalences between matchgate circuits, fermionic systems and 1D spin systems, the present work aims at gaining insight in the link between the physics of these systems and their computational capabilities. In this context one may pose a variety of interesting questions such as: ‘Does the presence of a quantum phase transition in the XY model leave any signature in the associated class of functions which can be computed by time-evolving such systems?’ The present work is also situated within such a program.

In the following we will characterize the family of matchgate-computable functions in full generality. We will find that this class precisely coincides with the class of linear threshold gates (LTGs) [14]. The latter is a fundamental family of functions that has been a topic of study since the 1960s and that plays an important role in numerous areas. LTGs occur in the study of neural networks where these functions serve as elementary models of neurons [15] and in circuit complexity theory [16]; cf. also e.g. [17] and references within for a number of recent investigations on LTGs. The existence of a connection between matchgates and LTGs may be considered surprising, since a priori there is no obvious relation between these two theories.

Below we state the contributions of this work in more precise terms. Here we highlight some particularly noteworthy aspects of our results.

An interesting phenomenon occurs when considering those functions that are matchgate-computable with high success probability. Since a generic matchgate circuit may be a rather complex object, one would expect that this class of functions has some nontrivial character as well. We will show that this intuition is incorrect: any function that is matchgate-computable with success probability greater than 3/4 is proved to be trivial in that any such function can only depend on a single bit of the input. The origin of this apparent paradox is the strong set of constraints that are placed on any circuit (matchgate or conventional) that is to compute the correct fun

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