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arXiv:quant-ph/0512170 · Physical Review A

Error-correcting codes for adiabatic quantum computation

Stephen P. Jordan, Edward Farhi, Peter W. Shor

Published Nov 14, 2006• 147 citations
arXivHTMLPDF

Abstract

Recently, there has been growing interest in using adiabatic quantum computation as an architecture for experimentally realizable quantum computers. One of the reasons for this is the idea that the energy gap should provide some inherent resistance to noise. It is now known that universal quantum computation can be achieved adiabatically using two-local Hamiltonians. The energy gap in these Hamiltonians scales as an inverse polynomial in the number of quantum gates being simulated. Here we present stabilizer codes which can be used to produce a constant energy gap against one-local and two-local noise. The corresponding fault-tolerant universal Hamiltonians are four-local and six-local, respectively, which are the optimal result achievable within this framework.

DOI
10.1103/physreva.74.052322

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Recently, there has been growing interest in using adiabatic quantum computation as an architecture for experimentally realizable quantum computers. One of the reasons for this is the idea that the energy gap should provide some inherent resistance to noise. It is now known that universal quantum computation can be achieved adiabatically using two-local Hamiltonians. The energy gap in these Hamiltonians scales as an inverse polynomial in the number of quantum gates being simulated. Here we present stabilizer codes which can be used to produce a constant energy gap against one-local and two-local noise. The corresponding fault-tolerant universal Hamiltonians are four-local and six-local, respectively, which are the optimal result achievable within this framework.
Adiabatic computation has been proposed as an architecture for quantum computation. In this project, we simulate an adiabatic computation de- signed to solve the MAX-2SAT problem, and study its behavior in the presence of noise. For this system, we implement a recently proposed error- correcting code (7) that is tolerant against 1-local errors. We show that the system can tolerate higher error rates using error-correction. trary to circuit model, the Hamiltonian of the sys- tem changes very slowly. One begins with a Hamil- tonian whose ground state is easy to construct, and gradually varies the Hamiltonian into one whose ground state encodes the solution to a computa- tional problem. It has been shown that the circuit model of quantum computers can be simulated by adiabatic computing (6). In this project we look at adiabatic computing in the presence of errors. We simulate a quantum adi- abatic algorithm designed to solve the NP-complete problem MAX-2SAT. We consider the in∞uence of quantum errors in the correctness probability of the algorithm. Recently, error-correcting codes speciflc for adiabatic quantum computation have been pro- posed. We implement these error-correcting codes for our particular instance of problem. We report the rate of error that these codes can tolerate.
We show that universal holonomic quantum computation can be achieved fault tolerantly by adiabatically deforming the gapped stabilizer Hamiltonian of the surface code, where quantum information is encoded in the degenerate ground space of the system Hamiltonian. We explicitly propose procedures to perform each logical operation, including logical state initialization, logical state measurement, logical controlled-not (cnot), state injection, distillation, etc. In particular, adiabatic braiding of different types of holes on the surface leads to a topologically protected, non-Abelian geometric logical cnot. Throughout the computation, quantum information is protected from both small perturbations and low-weight thermal excitations by a constant energy gap and is independent of the system size. Also, the Hamiltonian terms have weight at most four during the whole process. The effect of thermal error propagation is considered during the adiabatic code deformation. With the help of active error correction, this scheme is fault tolerant, in the sense that the computation time can be arbitrarily long for large-enough lattice size. It is shown that the frequency of error correction and the physical resources needed can be greatly reduced by the constant energy gap.
While adiabatic quantum computation (AQC) possesses some intrinsic robustness to noise, it is expected that a form of error control will be necessary for large scale computations. Error control ideas developed for circuit-model quantum computation do not transfer easily to the AQC model and to date there have been two main proposals to suppress errors during an AQC implementation: energy gap protection and dynamical decoupling. Here we show that these two methods are essentially equivalent and can be analyzed within the same formalism. We analyze the effectiveness of such error suppression techniques and identify critical constraints on the performance of error suppression in AQC, suggesting that error suppression by itself is insufficient for fault-tolerant, large-scale AQC and that a form of error correction is needed.
Fault-tolerant quantum computation allows quantum computations to be carried out while resisting unwanted noise. Several error-correcting codes have been developed to achieve this task, but none alone are capable of transversal universal quantum computation. This universality is highly desired and often achieved using techniques such as code concatenation, code switching, magic state distillation, or pieceable fault tolerance. Existing constructions can incur substantial overhead or rely on properties specific to particular codes. This work proposes implementing logical Clifford and T gates through novel, ancilla-mediated, pieceable interactions to construct a universal fault-tolerant quantum gate set. Our construction is deterministic, does not modify the underlying data codes or registers, and is generic over all data stabilizer codes. Thus, any single code becomes capable of universal quantum computation by leveraging helper codes in ancilla registers and mid-circuit measurements. Furthermore, since these logical gates are stabilizer code-generic, these implementations enable communication between heterogeneous stabilizer codes.
We show an equivalence relation between fault-tolerant circuits for a stabilizer code and fault-tolerant adiabatic processes for holonomic quantum computation (HQC), in the case where quantum information is encoded in the degenerate ground space of the system Hamiltonian. By this equivalence, we can systematically construct a fault-tolerant HQC scheme, which can geometrically implement a universal set of encoded quantum gates by adiabatically deforming the system Hamiltonian. During this process, quantum information is protected from low-weight thermal excitations by an energy gap that does not change with the problem size.