Universal Adiabatic Quantum Computation via the Space-Time Circuit-to-Hamiltonian Construction

Abstract

We show how to perform universal adiabatic quantum computation using a Hamiltonian which describes a set of particles with local interactions on a two-dimensional grid. A single parameter in the Hamiltonian is adiabatically changed as a function of time to simulate the quantum circuit. We bound the eigenvalue gap above the unique groundstate by mapping our model onto the ferromagnetic XXZ chain with kink boundary conditions; the gap of this spin chain was computed exactly by Koma and Nachtergaele using its q-deformed version of SU(2) symmetry. We also discuss a related time-independent Hamiltonian which was shown by Janzing to be capable of universal computation. We observe that in the limit of large system size, the time evolution is equivalent to the exactly solvable quantum walk on Young's lattice. [3,4]. Although the class of universal Hamiltonians originally considered (nearest neighbor interactions between six dimensional particles in two dimensions) is not practically viable, perturbation gadget techniques [5,6] were later used to massage it into simpler forms [7,8]. However, these techniques have the disadvantage of requiring impractically high variability in the coupling strengths which appear in the Hamiltonian (see, e.g., the analysis in [9]). Given this state of affairs, it is of interest to consider how to construct a universal adiabatic quantum computer with a simple Hamiltonian without using perturbative gadgets.An alternative type of circuit-to-Hamiltonian mapping which is conceptually distinct from the Feynman-Kitaev construction has been used by some authors [10][11][12][13][14][15][16]. In these works a quantum circuit is mapped to a Hamiltonian which acts on a Hilbert space with computational and "local" clock degrees of freedom associated with every qubit in the circuit. This idea was first explored by Margolus in 1989 [10], just four years after Feynman's celebrated paper on Hamiltonian computation [3]. Margolus showed how to simulate a one-dimensional cellular automaton by Schrödinger time evolution with a time-independent Hamiltonian. More recently, Janzing [11] presented a scheme for universal computation with a time-independent Hamiltonian. In reference [14] it was claimed that an approach along these lines can be used to perform universal adiabatic quantum computation; unfortunately, the analysis presented by Mizel et al. does not establish the claimed results. The local clock idea was developed further in the recent "space-time circuitto-Hamiltonian construction" and was used to prove that approximating the ground energy of a certain class of interacting particle systems is QMA-complete [16].Our main result is a new method which achieves efficient universal adiabatic quantum computation using the space-time circuit-to-Hamiltonian construction. The Hamiltonian we use describes a simple system of interacting particles which live on the edges of a two dimensional grid. To prove that the resulting algorithm is efficient we use a mapping from our Hamiltonian to the ferromagnetic XXZ model...

Decoherence in adiabatic quantum computation

“…However, what is true in the circuit model does not necessarily apply directly to other models of quantum computation, in particular the adiabatic model [2], in spite of the fact that the two models are computationally equivalent [3][4][5]. Thus, a commonly held belief, that a short single-qubit dephasing time necessarily implies quantum computational failure, should not be applied without first carefully specifying the computational model.…”

Adiabatic and Hamiltonian computing on a 2D lattice with simple two-qubit interactions

“…Initially, at the beginning of the computation, the particles are all residing on the left end of the grid, one on each site. Using nearest-neighbor interactions on the grid, one can embed a one-dimensional quantum circuit with nearest-neighbor interactions (see [20,25]). Each horizontal row then represents a single qubit wire of the one-dimensional quantum circuit.…”

“…Instead of using this rectangular grid one can imagine a line of particles hopping forward over a grid-a spatial execution of a one-dimensional quantum circuit. It was noted however in [20,25] that if the embedded quantum circuit is relatively small compared to m so that it can be embedded in the expanding region of the grid where particles are gradually added at the boundaries, then the forward motion of the particles in the Hamiltonian computation is very efficient. In essence the boundary condition imposed by this grid breaks the time-reversal symmetry of the computation: the string is more likely to move from the boundary to the bulk as the number of bulk string configurations is much larger.…”

“…The Hamiltonian and adiabatic computation that we propose bears the closest resemblance to the 2D grid model Hamiltonian in [25]: we show how mathematical results concerning this construction directly carry over to the results in this paper. The particle interactions in [25] (and the original [24]) require pairs of particles to jointly move or hop while updating their internal states: we expect that a reduction of such interactions to two-local interactions is inefficient, thus strongly favoring the novel construction in this paper. Two crucial steps give us this improvement over [25], namely the use of the railroad switch idea [8, 10, 28] to do a classical CNOT or Toffoli gate, and the use of 1st order perturbation theory to coordinate the joint movement of the particles so that they execute the computation jointly.A goal of this paper is to construct models for Hamiltonian and adiabatic quantum computation that are compatible with experimentally available couplings in a physical device such as 2D arrays of spin qubits in semiconducting quantum dots or superconducting qubits.…”

“…The particle interactions in [25] (and the original [24]) require pairs of particles to jointly move or hop while updating their internal states: we expect that a reduction of such interactions to two-local interactions is inefficient, thus strongly favoring the novel construction in this paper. Two crucial steps give us this improvement over [25], namely the use of the railroad switch idea [8, 10, 28] to do a classical CNOT or Toffoli gate, and the use of 1st order perturbation theory to coordinate the joint movement of the particles so that they execute the computation jointly.…”