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arXiv:1807.10379 · Physical Review A

Proof of efficient, parallelized, universal adiabatic quantum computation

Ari Mizel

Published Feb 12, 2019• 6 citations
arXivHTMLPDF

Abstract

We give a careful proof that a parallelized version of adiabatic quantum computation can efficiently simulate universal gate model quantum computation. The proof specifies an explicit parameter-dependent Hamiltonian $H(\ensuremath{\lambda})$ that is based on ground state quantum computation [Mizel et al., Phys. Rev. A. 63, 040302 (2001)]. We treat both a 1-dimensional configuration in which qubits on a line undergo nearest-neighbor 2-qubit gates and an all-to-all configuration in which every qubit undergoes 2-qubit gates with every other qubit in the system.

DOI
10.1103/physreva.99.022311

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We give a careful proof that a parallelized version of adiabatic quantum computation can efficiently simulate universal gate model quantum computation. The proof specifies an explicit parameter-dependent Hamiltonian H(\ensuremathλ)H(\ensuremath{\lambda})H(\ensuremathλ) that is based on ground state quantum computation [Mizel et al., Phys. Rev. A. 63, 040302 (2001)]. We treat both a 1-dimensional configuration in which qubits on a line undergo nearest-neighbor 2-qubit gates and an all-to-all configuration in which every qubit undergoes 2-qubit gates with every other qubit in the system.
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The model of adiabatic quantum computation is a relatively recent model of quantum computation that has attracted attention in the physics and computer science communities. We describe an efficient adiabatic simulation of any given quantum circuit. This implies that the adiabatic computation model and the standard circuit-based quantum computation model are polynomially equivalent. Our result can be extended to the physically realistic setting of particles arranged on a two-dimensional grid with nearest neighbor interactions. The equivalence between the models allows one to state the main open problems in quantum computation using well-studied mathematical objects such as eigenvectors and spectral gaps of Hamiltonians.
Quantum computation has revolutionary potential for speeding algorithms and for simulating quantum systems such as molecules. We report here a quantum computer design that performs universal quantum computation within a single non-degenerate ground state protected from decohering noise by an energy gap that we argue is system-size-independent. Closely analogous to a traditional electric circuit, it substantially changes the requirements for quantum computer construction, easing measurement, timing, and heating problems. Using the standard adiabatic condition, we present evidence that this design permits "quantum concurrent processing" distributing a quantum computation among extra qubits to perform a quantum algorithm of N gates in an amount of time that scales with the square root of N. One consequence of our work is a fixed gap version of adiabatic quantum computation, which several arguments hinted could be impossible.
We present two universal models of quantum computation with a time-independent, frustration-free Hamiltonian. The first construction uses 3-local (qubit) projectors and the second one requires only 2-local qubit-qutrit projectors. We build on Feynman’s Hamiltonian computer idea [R. Feynman, Optics News 11, 11 (1985)] and use a railroad-switch-type clock register. The resources required to simulate a quantum circuit with L gates in this model are O(L) small-dimensional quantum systems (qubits or qutrits), a time-independent Hamiltonian composed of O(L) local, constant norm, projector terms, the possibility to prepare computational basis product states, a running time O(L log2 L), and the possibility to measure a few qubits in the computational basis. Our models also give a simplified proof of the universality of 3-local adiabatic quantum computation.