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arXiv:1309.1534 · Physical Review A

Fault-tolerant scheme of holonomic quantum computation on stabilizer codes with robustness to low-weight thermal noise

Yi-Cong Zheng, Todd A. Brun

Published Mar 12, 2014• 13 citations
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Abstract

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.

DOI
10.1103/physreva.89.032317

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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.
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.
This paper generalizes and expands upon the work [O. Oreshkov, T. A. Brun, and D. A. Lidar, Phys. Rev. Lett. 102, 070502 (2009)] where we introduced a scheme for fault-tolerant holonomic quantum computation (HQC) on stabilizer codes. HQC is an all-geometric strategy based on non-Abelian adiabatic holonomies, which is known to be robust against various types of errors in the control parameters. The scheme we present shows that HQC is a scalable method of computation and opens the possibility for combining the benefits of error correction with the inherent resilience of the holonomic approach. We show that with the Bacon-Shor code the scheme can be implemented using Hamiltonian operators of weights 2 and 3.
Holonomic Quantum Computation (HQC) is an all-geometrical approach to quantum information processing. In the HQC strategy information is encoded in degenerate eigen-spaces of a parametric family of Hamiltonians. The computational network of unitary quantum gates is realized by driving adiabatically the Hamiltonian parameters along loops in a control manifold. By properly designing such loops the nontrivial curvature of the underlying bundle geometry gives rise to unitary transformations i.e., holonomies that implement the desired unitary transformations. Conditions necessary for universal QC are stated in terms of the curvature associated to the non-abelian gauge potential (connection) over the control manifold. In view of their geometrical nature the holonomic gates are robust against several kind of perturbations and imperfections. This fact along with the adiabatic fashion in which gates are performed makes in principle HQC an appealing way towards universal fault-tolerant QC.
In Chapter 16 it was shown how holonomic quantum computation (HQC) can be combined with the method of decoherence-free subspaces (DFSs), leading to passive protection against certain types of correlated errors. However, this is not enough for fault tolerance since other types of errors can accumulate detrimentally unless corrected. Scalability of HQC therefore requires going beyond that scheme, e.g., by combining the holonomic approach with active error correction. One way of combining HQC with active quantum error-correcting codes, which is similar to the way HQC is combined with DFSs, was also mentioned in Chapter 16. This approach, however, is not scalable since it requires Hamiltonians that commute with the stabilizer elements, and when the code increases in size, this necessitates couplings that become increasingly nonlocal. In this chapter, we will show how HQC can be made fault tolerant by combining it with the techniques for fault-tolerant quantum error correction (FTQEC) on stabilizer codes using Hamiltonians of finite locality. The fact that the holonomic method can be mapped directly to the circuit model allows us to construct procedures that resort almost entirely to these techniques. We will discuss an approach that makes use of the encoding already present in a stabilizer code and does not require additional qubits [O08, OBL08]. An alternative approach, which requires ancillary qubits for implementing transversal operations between qubits in the code, can be found in [O09].
The threshold theorem indicates that if errors are all local and their rates are below a certain threshold, it is possible to implement large scale quantum computation with arbitrarily small error based on active quantum error correction (QEC). However, most fault-tolerant quantum computation (FTQC) schemes require enormous overhead to achieve this rate by increasing either the concatenation levels for concatenated codes or the distance (hence the size) for topological codes. As a result, a logical qubit is usually encoded in more than thousands of physical qubits, even when the error rate is below the threshold. In this thesis, we try to reduce the resource overhead of FTQC. We first study a quantum computation scheme using clouds of atoms trapped on atomic chips. It suggests that this scheme is robust to noise to some extent and has potential to be scalable. Secondly, we propose a FTQC scheme of encoding many logical qubits into large code blocks. The error performance of such scheme is numerically studied. Preparation of various ancilla states necessary for computation is also explored. Thirdly, we explore the idea of introducing energy gap to suppress thermal noise on each qubit. We show that fault-tolerant holonomic quantum computation (HQC) can be implemented in stabilizer codes with the existence of such energy protection. Especially, we studied fault-tolerant HQC in surface codes in detail. This scheme opens the possibility for self-correcting quantum computation in 2D lattice.