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.
Matchgates are an especially multiflorous class of two-qubit nearest neighbour quantum gates, defined by a set of algebraic constraints. They occur for example in the theory of perfect matchings of graphs, non-interacting fermions, and one-dimensional spin chains. We show that the computational power of circuits of matchgates is equivalent to that of space-bounded quantum computation with unitary gates, with space restricted to being logarithmic in the width of the matchgate circuit. In particular, for the conventional setting of polynomial-sized (logarithmic-space generated) families of match…
Matchgates are a restricted set of two-qubit gates known to be classically simulable under particular conditions. Specifically, if a circuit consists only of nearest-neighbour matchgates, an efficient classical simulation is possible if either (i) the input is a computational basis state and the simulation requires computing probabilities of multi-qubit outcomes (including also adaptive measurements), or (ii) if the input is an arbitrary product state, but the output of the circuit consists of a single qubit. In this paper we extend these results to show that matchgates are classically simulab…
University of Augsburg（奥格斯堡大学）；Institut für Theoretische Physik, Universität zu Köln（科隆大学理论物理研究所）；Barcelona Supercomputing Center（巴塞罗那超级计算中心）；Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）
In Valiant's matchgate theory, 2-input 2-output matchgates are 4x4 matrices that satisfy ten so-called matchgate identities. We prove that the set of all such matchgates (including non-unitary and non-invertible ones) coincides with the topological closure of the set of all matrices obtained as exponentials of linear combinations of the 2-qubit Jordan-Wigner (JW) operators and their quadratic products, extending a previous result of Knill. In Valiant's theory, outputs of matchgate circuits can be classically computed in poly-time. Via the JW formalism, Terhal & DiVincenzo and Knill established…
Magic states were introduced in the context of Clifford circuits as a resource that elevates classically simulatable computations to quantum universal capability, while maintaining the same gate set. Here we study magic states in the context of matchgate (MG) circuits, where the notion becomes more subtle, as MGs are subject to locality constraints and also the SWAP gate is not available. Nevertheless a similar picture of gate-gadget constructions applies, and we show that every pure fermionic state which is non-Gaussian, i.e. which cannot be generated by MGs from a computational basis state, …
We propose a method to reliably and efficiently extract the fidelity of many-qubit quantum circuits composed of continuously parametrized two-qubit gates called matchgates. This method, which we call matchgate benchmarking, relies on advanced techniques from randomized benchmarking as well as insights from the representation theory of matchgate circuits. We argue the formal correctness and scalability of the protocol, and moreover deploy it to estimate the performance of matchgate circuits generated by two-qubit XY spin interactions on a quantum processor.
We study the classical simulation complexity in both the weak and strong senses, of matchgate (MG) computations supplemented with all combinations of settings involving inclusion of intermediate adaptive or nonadaptive computational basis measurements, product state or magic and general entangled state inputs, and single- or multi-line outputs. We find a striking parallel to known results for Clifford circuits, after some rebranding of resources. We also give bounds on the amount of classical simulation effort required in case of limited access intermediate measurements and entangled inputs. I…
University of Chicago（芝加哥大学）；Center for Theoretical Physics, Polish Academy of Sciences（波兰科学院理论物理中心）；University of Michigan（密歇根大学）；IBM Quantum, IBM T.J. Watson Research Center（IBM量子，IBM托马斯·J·沃森研究中心）
The theory of matchgates is of interest in various areas in physics and computer science. Matchgates occur in e.g. 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 which appears in several fields, such as the study of neural networks. Using …
Matchgates are a restricted set of two-qubit gates known to be classically simulable when acting on nearest-neighbor qubits on a path, but universal for quantum computation when the qubits are arranged on certain other graphs. Here we characterize the power of matchgates acting on arbitrary graphs. Specifically, we show that they are universal on any connected graph other than a path or a cycle, and that they are classically simulable on a cycle. We also prove the same dichotomy for the XY interaction, a proper subset of matchgates related to some implementations of quantum computing.
In the circuit model, quantum computers rely on the availability of a universal quantum gate set. A particularly intriguing example is a set of two-qubit only gates: matchgates, along with SWAP (the exchange of two qubits). In this paper, we show a simple decomposition of arbitrary matchgates into better known elementary gates, and implement a matchgate in a linear-optics experiment using single photons. The gate performance was fully characterized via quantum process tomography. Moreover, we represent the resulting reconstructed quantum process in a novel way, as a fidelity map in the space o…
Matchgates are a group of two-qubit gates associated with free fermions. They are classically simulatable if restricted to act between nearest neighbors on a one-dimensional chain, but become universal for quantum computation with longer-range interactions. We describe various alternative geometries with nearest-neighbor interactions that result in universal quantum computation with matchgates only, including subtle departures from the chain. Our results pave the way for new quantum computer architectures that rely solely on the simple interactions associated with matchgates.
Matchgates are a family of two-qubit gates associated with noninteracting fermions. They are classically simulatable if acting only on nearest neighbors, but become universal for quantum computation if we relax this restriction or use SWAP gates [Jozsa and Miyake, Proc. R. Soc. A 464, 3089 (2008)]. We generalize this result by proving that any nonmatchgate parity-preserving unitary is capable of extending the computational power of matchgates into universal quantum computation. We identify the single local invariant of parity-preserving unitaries responsible for this, and discuss related resul…
Complexity separations for optimal matchgate-Clifford synthesis
Barcelona Supercomputing Center（巴塞罗那超级计算中心）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Johannes Kepler University（约翰内斯·开普勒大学）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Princeton University（普林斯顿大学）
Dynamic simulation of materials is a promising application for near-term quantum computers. Current algorithms for Hamiltonian simulation, however, produce circuits that grow in depth with increasing simulation time, limiting feasible simulations to short-time dynamics. Here, we present a method for generating circuits that are constant in depth with increasing simulation time for a subset of one-dimensional materials Hamiltonians, thereby enabling simulations out to arbitrarily long times. Furthermore, by removing the effective limit on the number of feasibly simulatable time-steps, the const…
Fermionic linear optics is efficiently classically simulatable. Here it is shown that the set of states achievable with fermionic linear optics and particle measurements is the closure of a low dimensional Lie group. The weakness of fermionic linear optics and measurements can therefore be explained and contrasted with the strength of bosonic linear optics with particle measurements. An analysis of fermionic linear optics is used to show that the two-qubit matchgates and the simulatable matchcircuits introduced by Valiant generate a monoid of extended fermionic linear optics operators. A usefu…
We describe a simple formalism for generating classes of quantum circuits that are classically efficiently simulatable and show that the efficient simulation of Clifford circuits (Gottesman-Knill theorem) and of matchgate circuits (Valiant's theorem) appear as two special cases. Viewing these simulatable classes as subsets of the space of all quantum computations, we may consider minimal extensions that suffice to regain full quantum computational power, which provides an approach to exploring the efficacy of quantum over classical computation.
Shortened abstract: In this thesis, I study two restricted models of quantum computing related to free identical particles. Free fermions correspond to a set of two-qubit gates known as matchgates. Matchgates are classically simulable when acting on nearest neighbors on a path, but universal for quantum computing when acting on distant qubits or when SWAP gates are available. I generalize these results in two ways. First, I show that SWAP is only one in a large family of gates that uplift matchgates to quantum universality. In fact, I show that the set of all matchgates plus any nonmatchgate p…
We test the principle of majorization [J. I. Latorre and M. A. Martin-Delgado, Phys. Rev. A 66, 022305 (2002)] in random circuits. Three classes of circuits were considered: (i) universal, (ii) classically simulatable, and (iii) neither universal nor classically simulatable. The studied families are: {CNOT, H, T}, {CNOT, H, NOT}, {CNOT, H, S} (Clifford), matchgates, and IQP (instantaneous quantum polynomial-time). We verified that all the families of circuits satisfy on average the principle of decreasing majorization. In most cases the asymptotic state (number of gates going to infinity) beha…
Compressible models extend the domain of simulable systems in quantum computers, but little is known about their precise limits of applicability. Using the theory of compressible matchgate circuits, we identify a class of quadratic fermionic Hamiltonians that can be simulated in compressed space. In particular, for systems of $n$ orbitals encoded to 2-local qubit models with nearest neighbour interactions, the ground state energy can be evaluated with $O\left(\log n\right)$ sets of measurements, independently of the number of dimensions in which the $n$ sites are arranged. We also provide an e…
It remains an open question whether the apparent additional power of quantum computation derives inherently from quantum mechanics, or merely from the flexibility obtained by "lifting" Boolean functions to linear operators and evaluating their composition cleverly. Holographic algorithms provide a useful avenue for exploring this question. We describe a new, simplified construction of holographic algorithms in terms of Pfaffian circuits. Novel proofs of some key results are provided, and we extend the approach of [34] to nonsymmetric, odd, and homogenized signatures, circuits, and various mode…
Quantum simulations are bound to be one of the main applications of near-term quantum computers. Quantum chemistry and condensed matter physics are expected to benefit from these technological developments. Several quantum simulation methods are known to prepare a state on a quantum computer and measure the desired observables. The most resource economic procedure is the variational quantum eigensolver (VQE), which has traditionally employed unitary coupled cluster as the ansatz to approximate ground states of many-body fermionic Hamiltonians. A significant caveat of the method is that the ini…
We investigate the boundary between classical and quantum computational power. This work consists of two parts. First we develop new classical simulation algorithms that are centered on sampling methods. Using these techniques we generate new classes of classically simulatable quantum circuits where standard techniques relying on the exact computation of measurement probabilities fail to provide efficient simulations. For example, we show how various concatenations of matchgate, Toffoli, Clifford, bounded-depth, Fourier transform and other circuits are classically simulatable. We also prove th…
We extend the concept of Anderson localization, the confinement of quantum information in a spatially irregular potential, to quantum circuits. Considering matchgate circuits, generated by time-dependent spin-1/2 XY Hamiltonians, we give an analytic formula for the out-of-time-ordered correlator of a local observable, and show that it can be efficiently evaluated by a classical computer even when the explicit Heisenberg time evolution cannot. Because this quantity bounds the average error incurred by truncating the evolution to a spatially limited region, we demonstrate dynamical localization …
