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The Hubbard Model for Universal Quantum Computation

Jiawei Ji

Published Jan 24, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.11575/prism/36111

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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.
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.
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.
Matchgate circuits realize free-fermion dynamics: they are efficiently classically simulable, yet cannot on their own generate the generic randomness required for universal computation or unitary design formation. We study a controlled route beyond this integrable limit by doping matchgate circuits with non-Gaussian gates-physically, the injection of fermionic interactions into an otherwise free system. Using the matchgate commutant framework, we obtain analytic control over unitary 222-design formation. For globally scrambled dynamics, the design problem maps exactly onto a classical birth-death Markov chain with an Ornstein-Uhlenbeck continuum limit, recasting the emergence of quantum randomness in terms of spectral gaps and mixing times and yielding rigorous bounds on the number of non-Gaussian gates needed for approximate 222-designs. These bounds hold for a broad class of parity-preserving non-Gaussian gates, independently of microscopic details, with numerics indicating that the same mechanism governs higher-order designs. Used as local building blocks in a glued-circuit architecture, they yield approximate parity-preserving 222-designs in polylogarithmic depth with a sparse non-Gaussian gate count, with implications for Page-like entanglement growth and fermionic classical-shadow protocols. Finally, locality reshapes this picture: in local brickwork dynamics, design formation is diffusion-limited and far slower. Our results establish doped matchgate circuits as a controlled, analytically tractable route from free fermions to interaction-generated quantum designs.
Because Majorana zero modes store quantum information nonlocally, they are protected from noise, and have been proposed as a building block for a quantum computer. We show how to use the same protection from noise to implement universal fermionic quantum computation. Our architecture requires only two Majorana modes to encode a fermionic quantum degree of freedom, compared to alternative implementations which require a minimum of four Majorana modes for a spin quantum degree of freedom. The fermionic degrees of freedom support both unitary coupled cluster variational quantum eigensolver and quantum phase estimation algorithms, proposed for quantum chemistry simulations. Because we avoid the Jordan-Wigner transformation, our scheme has a lower overhead for implementing both of these algorithms, allowing for simulation of the Trotterized Hubbard Hamiltonian in O(1) time per unitary step. We finally demonstrate magic state distillation in our fermionic architecture, giving a universal set of topologically protected fermionic quantum gates.
Topological quantum computation by way of braiding of Majorana fermions is not universal quantum computation. There are several attempts to make universal quantum computation by introducing some additional quantum gates or quantum states. However, there is a serious problem in embedding an MMM-qubit quantum gate in the NNN-qubit system with N>MN>MN>M. This is an inherent problem to the Majorana system, where quantum gates for logical qubits become nonlocal in terms of physical qubits in general because braiding operations preserve the fermion parity. For instance, the CZ gate could not be embedded in the three-qubit system. We overcome this embedding problem by introducing 2(N+1)2(N+1)2(N+1)-body interactions of Majorana fermions in the NNN-qubit system. A universal set of quantum gates is constructed for NNN logical qubits, leading to topological-nontopological hybrid universal quantum computation. It would be more robust than conventional universal quantum computation because the quantum gates generated by braiding are topologically protected.