Surface codes can protect quantum information stored in qubits from local errors as long as the per-operation error rate is below a certain threshold. Here we propose holonomic surface codes by harnessing the quantum holonomy of the system. In our scheme, the holonomic gates are built via auxiliary qubits rather than the auxiliary levels in multilevel systems used in conventional holonomic quantum computation. The key advantage of our approach is that the auxiliary qubits are in their ground state before and after each gate operation, so they are not involved in the operation cycles of surface codes. This provides an advantageous way to implement surface codes for fault-tolerant quantum computation.
Surface codes can protect quantum information stored in qubits from local errors as long as the per-operation error rate is below a certain threshold. Here we propose holonomic surface codes by harnessing the quantum holonomy of the system. In our scheme, the holonomic gates are built via auxiliary qubits rather than the auxiliary levels in multilevel systems used in conventional holonomic quantum computation. The key advantage of our approach is that the auxiliary qubits are in their ground state before and after each gate operation, so they are not involved in the operation cycles of surface codes. This provides an advantageous way to implement surface codes for fault-tolerant quantum computation.
It is well known that surface codes are favorable candidates toward realizing large-scale fault-tolerant quantum computation. While methods to reduce errors by surface codes have been well explored, it is still challenging to achieve small-enough errors as stated by the threshold theorem in practical experimental settings. This prompts the integration of surface codes and protected quantum operations to further mitigate the negative effect of errors, creating the challenge of executing protected surface codes. In this work, we propose one scheme to perform holonomic quantum operations in a physical system by use of auxiliary qubits. The system Hamiltonian describes a quantum spin model with adjustable coupling strength. Various noncommutable single-qubit and nontrivial two-qubit holonomic gates are applicable on the basis of the interaction between the auxiliary qubits and physical qubits. By considering a two-dimensional lattice of qubits, we explore the execution of surface codes. The utilization rate of physical qubits as data qubits is about $24.7\mathrm{%}$ in executing the protected surface codes. Our scheme is based on an experimentally achievable Hamiltonian and thus it brings us closer to realizing protected quantum operations on quantum-error-correction codes with a modest resource.
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.
Quantum computers are far more error-prone than their classical counterparts. Therefore, to build a quantum computer capable of running large-scale quantum algorithms, we must use the techniques of quantum error correction to ensure that the computer produces the correct output even when its components are unreliable. However, the resource requirements of building such a fault-tolerant quantum computer are currently prohibitive. Here, we examine the utility of using three-dimensional (3D) surface codes in a fault-tolerant quantum computer. This family of topological error-correcting codes is a generalization of the well-known 2D surface code to three spatial dimensions. We show that certain 3D surface codes have a transversal logical non-Clifford gate. In a quantum computing architecture, a non-Clifford gate is required to achieve computational universality. Transversal gates do not entangle qubits in different codes, so they are naturally fault tolerant because they do not spread errors. Next, we consider the problem of decoding 3D surface codes. In a quantum error-correcting code, we cannot observe the qubits directly, so we measure parity-check operators to gain information about the state of the code. Decoding is the problem of estimating what error has occurred given a list of unsatisfied parity checks. We observe that 3D surface codes offer asymmetric protection against bit-flip and phase-flip errors, but in both cases, we find that a threshold error rate exists below which we can suppress logical errors by increasing the size of the code. We use our results about logical gates and decoding to propose two fault-tolerant quantum computing architectures that utilize 3D surface codes. Finally, we compare the resource requirements of our architectures with the requirements of leading quantum computing architectures based on topological codes. We find that one of our architectures may be competitive with the leading architectures, depending on the properties of the physical systems used to build the qubits.
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.
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].
Surface codes represent a promising route towards universal fault-tolerant quantum computation, allowing for the protection of stored quantum states against many types of errors. This theoretical work shows that, in addition, a significantly enhanced versatility in quantum information processing is expected when qubits are encoded into the Majorana bound states in topological superconductor-semiconductor heterostructures. In view of recent experimental evidence for Majorana states in such heterostructures, a two-dimensional network of interacting Majorana states thus may feature key advantages towards the long-term goal of a functional quantum computer. The authors discuss how topologically protected logical qubits in this Majorana surface code architecture can be defined, initialized, manipulated, and read out. The physical ingredients needed to implement such operations are familiar from topologically trivial quantum devices. In particular, by employing quantum interference terms in conductance measurements, composite single-electron pumping protocols, and gate-tunable tunnel barriers, the full set of gates required for universal quantum computation can be implemented.