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arXiv:1107.2697 · Physical Review Letters

Nonperturbative Gadget for Topological Quantum Codes

Samuel A. Ocko, Beni Yoshida

Published Dec 14, 2011• 22 citations
arXivHTMLPDF

Abstract

Many-body entangled systems, in particular topologically ordered spin systems proposed as resources for quantum information processing tasks, often involve highly nonlocal interaction terms. While one may approximate such systems through two-body interactions perturbatively, these approaches have a number of drawbacks in practice. In this Letter, we propose a scheme to simulate many-body spin Hamiltonians with two-body Hamiltonians nonperturbatively. Unlike previous approaches, our Hamiltonians are not only exactly solvable with exact ground state degeneracy, but also support completely localized quasiparticle excitations, which are ideal for quantum information processing tasks. Our construction is limited to simulating the toric code and quantum double models, but generalizations to other nonlocal spin Hamiltonians may be possible.

DOI
10.1103/physrevlett.107.250502

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Many-body entangled systems, in particular topologically ordered spin systems proposed as resources for quantum information processing tasks, often involve highly nonlocal interaction terms. While one may approximate such systems through two-body interactions perturbatively, these approaches have a number of drawbacks in practice. In this Letter, we propose a scheme to simulate many-body spin Hamiltonians with two-body Hamiltonians nonperturbatively. Unlike previous approaches, our Hamiltonians are not only exactly solvable with exact ground state degeneracy, but also support completely localized quasiparticle excitations, which are ideal for quantum information processing tasks. Our construction is limited to simulating the toric code and quantum double models, but generalizations to other nonlocal spin Hamiltonians may be possible.
We develop and investigate several quantum many-body spin models of use for topological quantum information processing and storage. These models fall into two categories: those that are designed to be more realistic than alternative models with similar phenomenology, and those that are designed to have richer phenomenology than related models. In the first category, we present a procedure to obtain the Hamiltonians of the toric code and Kitaev quantum double models as the perturbative low-energy limits of entirely two-body Hamiltonians. This construction reproduces the target models' behavior using only couplings which are natural in terms of the original Hamiltonians. As an extension of this work, we construct parent Hamiltonians involving only local 2-body interactions for a broad class of Projected Entangled Pair States (PEPS). We define a perturbative Hamiltonian with a finite order low energy effective Hamiltonian that is a gapped, frustration-free parent Hamiltonian for an encoded version of a desired PEPS. For topologically ordered PEPS, the ground space of the low energy effective Hamiltonian is shown to be in the same phase as the desired state to all orders of perturbation theory. We then move on to define models that generalize the phenomenology of several well-known systems. We first define generalized cluster states based on finite group algebras, and investigate properties of these states including their PEPS representations, global symmetries, relationship to the Kitaev quantum double models, and possible applications. Finally, we propose a generalization of the color codes based on finite groups. For non-Abelian groups, the resulting model supports non-Abelian anyonic quasiparticles and topological order. We examine the properties of these models such as their relationship to Kitaev quantum double models, quasiparticle spectrum, and boundary structure.
The need for Hamiltonians with many-body interactions arises in various applications of quantum computing. However, interactions beyond two-body are difficult to realize experimentally. Perturbative gadgets were introduced to obtain arbitrary many-body effective interactions using Hamiltonians with at most two-body interactions. Although valid for arbitrary kkk-body interactions, their use is limited to small kkk because the strength of interaction is kthk\mathrm{th}kth order in perturbation theory. In this paper we develop a nonperturbative technique for obtaining effective kkk-body interactions using Hamiltonians consisting of at most lll-body interactions with l<kl<kl<k. The idea is to first apply a unitary transformation on the system plus ancilla qubits (possibly using a gate-based device), then evolve with a new Hamiltonian which is more local than the original one (using an analog device), and finally reverse the unitary transformation. The net effect of this procedure is shown to be equivalent to evolving the system with the original nonlocal Hamiltonian. This technique does not suffer from the aforementioned shortcoming of perturbative methods and requires only one ancilla qubit for each kkk-body interaction irrespective of the value of kkk. It works best for Hamiltonians with a few many-body interactions involving a large number of qubits and can be used together with perturbative gadgets to embed Hamiltonians of considerable complexity in proper subspaces of two-local Hamiltonians. We describe how our technique can be implemented in a hybrid (gate-based and adiabatic) as well as solely adiabatic quantum computing scheme.
Abstract Continuous-time quantum hardware implementations generally lack the native capability to implement high-order terms that would facilitate efficient compilation of quantum algorithms. This limitation has, in part, motivated the development of perturbative gadgets—multi-qubit constructions used to effect a desired Hamiltonian using engineered low-energy subspaces of a larger system constructed using simpler, usually two-body, primitives. In this work, we demonstrate how a class of non-perturbative gadgets can produce high-order multi-body interactions by taking advantage of the odd–even properties of topological defect subspaces. The simplest example uses domain-wall defects in an effective Ising spin chain with linear connectivity and three-body couplings, alongside three- or five-body driving terms depending on the intended use. We demonstrate a version of a gadget which can perform an encoded bit-flip operation on a minor embedding chain, an important task to mitigate the limitations of quasi-two-dimensional (also sometimes called quasi-planar) topology. Although this will be the main focus of the paper due to conceptual simplicity, there exist systems constructed with only two-body couplings where the boundaries determine whether there are an odd or even number of defects, namely ice-like systems which may yield more complex gadget-like constructions.
We present here a procedure to obtain the Hamiltonians of the toric code and Kitaev quantum double models as the low-energy limits of entirely two-body Hamiltonians. Our construction makes use of a new type of perturbation gadget based on error-detecting subsystem codes. The procedure is motivated by a projected entangled pair states (PEPS) description of the target models, and reproduces the target models' behavior using only couplings that are natural in terms of the original Hamiltonians. This allows our construction to capture the symmetries of the target models.
We propose a new quantum simulation method for simulating N-body interactions, which are tensor products of N Pauli operators, in an analytically exact manner. This method iteratively attaches many two-body interactions on one two-body interaction to simulate an N-body interaction. Those controlled two-body interactions can be untunable and act only on neighboring spins. The strength difference between controlled and target Hamiltonians is normally not more than one order of magnitude. This simulation is theoretically error-free, and errors due to experimental imperfections are ignorable. A major obstacle to simulating the toric code model and modified toric codes used in topological quantum computation is to simulate N-body interactions. We employ the new quantum simulation method to solve this issue and thus simulate the toric code model and its modifications.
We construct parent Hamiltonians involving only local 2-body interactions for a broad class of projected entangled pair states (PEPS). Making use of perturbation gadget techniques, we define a perturbative Hamiltonian acting on the virtual PEPS space with a finite order low energy effective Hamiltonian that is a gapped, frustration-free parent Hamiltonian for an encoded version of a desired PEPS. For topologically ordered PEPS, the ground space of the low energy effective Hamiltonian is shown to be in the same phase as the desired state to all orders of perturbation theory. An encoded parent Hamiltonian for the double semion string net ground state is explicitly constructed as a concrete example.