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Physical Review A

Extending matchgates to universal quantum computation via the Hubbard model

Jia-Wei Ji, David L. Feder

Published Nov 19, 2019• 3 citations
Publisher

Abstract

Quantum circuits solely comprising matchgates can perform nontrivial (but nonuniversal) quantum algorithms. Because matchgates can be mapped to noninteracting fermions, these circuits can be efficiently simulated on a classical computer. Universal quantum computation is attainable by adding any nonmatchgate parity-preserving gate, from which one may infer that interacting fermions are natural candidates for universal quantum computation. We consider the quantum computational power of fermions hopping on a one-dimensional double-well lattice within the context of matchgates. In particular, we show that universal quantum computation can be implemented using spinless (spin-polarized) fermions and nearest-neighbor interactions, as well as with spin-half fermions with on-site interactions (i.e., the Hubbard model). We suggest that these schemes are currently within reach in the context of ultracold atomic gases.

DOI
10.1103/physreva.100.052324

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Quantum circuits solely comprising matchgates can perform nontrivial (but nonuniversal) quantum algorithms. Because matchgates can be mapped to noninteracting fermions, these circuits can be efficiently simulated on a classical computer. Universal quantum computation is attainable by adding any nonmatchgate parity-preserving gate, from which one may infer that interacting fermions are natural candidates for universal quantum computation. We consider the quantum computational power of fermions hopping on a one-dimensional double-well lattice within the context of matchgates. In particular, we show that universal quantum computation can be implemented using spinless (spin-polarized) fermions and nearest-neighbor interactions, as well as with spin-half fermions with on-site interactions (i.e., the Hubbard model). We suggest that these schemes are currently within reach in the context of ultracold atomic gases.
Quantum circuits based only on matchgates are able to perform non-trivial (but not universal) quantum algorithms. Because matchgates can be mapped to non-interacting fermions, these circuits can be efficiently simulated on a classical computer. One can perform universal quantum computation by adding any non-matchgate parity-preserving gate, implying that interacting fermions are natural candidates for universal quantum computation. Most work to date has focused on Majorana fermions, which are difficult to realize and manipulate in the laboratory, despite the advantage of topologically protecting quantum information. We instead show that universal quantum computation can be implemented using interacting spinless (spin-polarized) fermions and further propose a scheme for achieving universal quantum computation with the Hubbard model, which may be realized in the laboratory based on current experimental techniques.
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 results in the context of fermionic systems.
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.
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 parity-preserving two-qubit gate is universal, and interpret this fact in terms of local invariants of two-qubit gates. Second, I investigate the power of matchgates in arbitrary connectivity graphs, showing they are universal on any connected graph other than a path or a cycle, and classically simulable on a cycle. I also prove the same dichotomy for the XY interaction. Free bosons give rise to a model known as BosonSampling. BosonSampling consists of (i) preparing a Fock state of n photons, (ii) interfering these photons in an m-mode linear interferometer, and (iii) measuring the output in the Fock basis. Sampling approximately from the resulting distribution should be classically hard, under reasonable complexity assumptions. Here I show that exact BosonSampling remains hard even if the linear-optical circuit has constant depth. I also report several experiments where three-photon interference was observed in integrated interferometers of various sizes, providing some of the first implementations of BosonSampling in this regime. The experiments also focus on the bosonic bunching behavior and on validation of BosonSampling devices. This thesis contains descriptions of the numerical analyses done on the experimental data, omitted from the corresponding publications.
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.
We present a universal framework to harness fermionic ultracold atom platforms for quantum simulation, showing how variational algorithms on existing hardware can simulate many-body systems well beyond the hardware's native Hamiltonian. Our analysis provides evidence that one can quantum simulate the ground-state properties of a broad class of gapless target Hamiltonians of local observables in a quantum evolution time that grows polynomially with the inverse relative error, T∼O(poly(1/ε))T\sim O(\mathrm{poly}(1/ε))T∼O(poly(1/ε)) up to logarithmic corrections, offering an exponential speedup over na{ï}ve classical algorithms such as exact diagonalization. We provide numerical evidence and theoretical argument that this holds for energy density, density-density, and spin-spin correlations in three qualitatively distinct models -- the repulsive Hubbard model; a Hubbard model augmented with nearest-neighbor attractive interactions, which introduces the phenomenon of pairing; and the Hofstadter-Hubbard model, which introduces a gauge field and fractional quantum Hall physics. This work demonstrates quantum simulation using current fermionic platforms far beyond the models natively implemented in the hardware.