Measurement-Based Quantum Computer in the Gapped Ground State of a Two-Body Hamiltonian

Abstract

We propose a scheme for a ground-code measurement-based quantum computer, which enjoys two major advantages. First, every logical qubit is encoded in the gapped degenerate ground subspace of a spin-1 chain with nearest-neighbor two-body interactions, so that it equips built-in robustness against noise. Second, computation is processed by single-spin measurements along multiple chains dynamically coupled on demand, so as to keep teleporting only logical information into a gapprotected ground state of the residual chains after the interactions with spins to be measured are turned off. We describe implementations using trapped atoms or polar molecules in an optical lattice, where the gap is expected to be as large as 0.2 kHz or 4.8 kHz respectively. Introduction.-Reliable quantum computers require hardware with low error rates and sufficient resources to perform software-based error correction. One appealing approach to reduce the massive overhead for error correction is to process quantum information in the gapped ground states of some many-body interaction. This is the tactic used in topological quantum computation and adiabatic quantum computation. Yet, the hardware demands for the former are substantial, and the fault tolerance of the later, especially when restricted to two-body interactions, is unclear [1]. On the other hand, measurement-based quantum computation (MQC), in particular one-way computation on the 2D cluster state [2], runs by beginning with a highly entangled state dynamically generated from nearest-neighbor two-body interactions and performing computation by only single-qubit measurements and feed forward of their outcomes. However, its bare implementation may suffer decoherence of physical qubits waiting for their round of measurements in the far future, and that severely damages a prominent capability to parallelize computation. Although its fault-tolerant method by error correction has been well established [3], it is clearly advantageous if some gap-protection is provided on the hardware level.

Quantum computational capability of a 2D valence bond solid phase

“…As originally proposed in Ref. [11] as the ground-code version of MQC using coupled 1D AKLT states, if we are capable of adiabatically turning off a two-body interacting term only to the spin to be measured selectively -importantly that still does not have to create new entanglement -we could indeed make the Hamiltonian in the remaining bulk coexist without interfering the MQC protocol for quantum computation. Then, the Hamiltonian with a (conjectured) gap would be able to contribute to a passive protection of the logical information stored in the degenerate ground state, by providing some robustness against local noises, in compared to a scenario where the bulk Hamiltonian is absent.…”

“…This seems to be the reason why MQC on the 2D AKLT state has been an open question in a long time, although the AKLT state by the 1D spin-1 chain was shown in Ref. [11] to be capable of simulating a single quantum wire of MQC.…”

“…Notably, however, most known examples considered so far, including additionally those e.g., in Refs. [11][12][13][14][15][16], are constructed to have such a convenient yet artificial property -as often referred as one of peculiar properties of the correlations of the 2D cluster state -that it is possible to decouple deterministically (by measurements of only neighboring sites) a 1D-chain structure that encodes the direction of a simulated time as a quantum logical wire of the quantum circuit model. This peculiarity is said to be artifact of another less realistic feature of the 2D cluster state in that it cannot be the exact ground state of any two-body Hamiltonian of spin 1 2 's [17,18].…”

Strategies for measurement-based quantum computation with cluster states transformed by stochastic local operations and classical communication

“…One might expect that one or more of these properties would be necessarily satisfied by any universal resource state. This has turned out not to be the case; several authors in recent years have identified resources that differ materially from the cluster states [4,5,[9][10][11][12][13][14].…”

Universal measurement-based quantum computation with spin-2 Affleck-Kennedy-Lieb-Tasaki states

“…Moreover, they are unique ground states of two-body interacting Hamiltonians with suitable boundary conditions. Even though quantum computational universality in MBQC requires at least a 2D structure, the results on the 1D spin-1 AKLT state for theoretically and experimentally simulating one-qubit gates [11,27,28] prompted the quest of universality in 2D AKLT states. However, the capability of full quantum computational universality was only established recently in the spin-3/2 AKLT state on the honeycome lattice [29,30] and later on some other trivalent lattices [31], as well as some lattices that host spin-2 and other lower spin (such as spin-3/2 or spin-1) hybrid AKLT states [32].…”