We present a family of algorithms, combining real-space renormalization methods and belief propagation, to estimate the free energy of a topologically ordered system in the presence of defects. Such an algorithm is needed to preserve the quantum information stored in the ground space of a topologically ordered system and to decode topological error-correcting codes. For a system of linear size , our algorithm runs in time log compared to 6 needed for the minimum-weight perfect matching algorithm previously used in this context and achieves a higher depolarizing error threshold.Topologically ordered phases of matter can be used to store and process quantum information in an inherently robust way [1][2][3][4][5]. The ground state degeneracy depends on the topology of the system. Quantum information stored in this ground state manifold is protected from local perturbations because virtual transitions require an order in perturbation proportional to the linear size of the system.At finite temperature, thermal excitations create pairs (or larger sets) of particles of finite mass. These particles are not confined and can freely diffuse on the surface. Bringing all the particles to fuse in pairs returns the system into a ground state. This ground state will correspond to the original one only if the world-line of the particles has a trivial homology: otherwise the particles have generated a topological transformation that corrupts the information stored in the ground space. Since particles appear in a finite density (e − mass k B T ) at any nonzero temperature, this information corruption happens on a time-scale independent of the system size [6].To store the information for longer times, it becomes necessary to keep track of the thermal particles and have accurate knowledge of their world-line homology. This is possible if the locations of the particles are measured repeatedly on a time-scale shorter or comparable to their diffusion rate. It is also necessary to process the information gathered from these measurements rapidly, i.e., to infer the world-line homology from knowledge of the particle configuration at discrete times.For some models, such as color codes [7], there is no known efficient algorithm that can infer the particles' world-line homology. Other models, such as Kitaev's code, admit an efficient algorithm-the minimum-weight perfect matching algorithm of Edmonds [8] (PMA)-that solves this problem in time proportional to 6 where is the linear size of the system [4,9]. With such a scaling, it is not possible to handle lattices of more than a few hundred sites long in any reasonable time, ruling out any practical application in this context. Additionally, this algorithm is sub-optimal. As we show below, decoding a topological code reduces to minimizing the free-energy over all homology classes of the system. The PMA minimizes the energy instead, which can lead to additional corruption of the information.In this Letter, we present a real-space renormalization group (RG) algorithm to accomplish this task in t...

Steane's 7-qubit quantum error-correcting code admits a set of fault-tolerant gates that generate the Clifford group, which in itself is not universal for quantum computation. The 15-qubit Reed-Muller code also does not admit a universal fault-tolerant gate set but possesses fault-tolerant T and control-control-Z gates. Combined with the Clifford group, either of these two gates generates a universal set. Here, we combine these two features by demonstrating how to fault-tolerantly convert between these two codes, providing a new method to realize universal fault-tolerant quantum computation. One interpretation of our result is that both codes correspond to the same subsystem code in different gauges. Our scheme extends to the entire family of quantum Reed-Muller codes.

Topological phases can be defined in terms of local equivalence: two systems are in the same topological phase if it is possible to transform one into the other by a local reorganization of its degrees of freedom. The classification of topological phases therefore amounts to the classification of long-range entanglement. Such local transformation could result, for instance, from the adiabatic continuation of one system's Hamiltonian to the other. Here, we use this definition to study the topological phase of translationally invariant stabilizer codes in two spatial dimensions, and show that they all belong to one universal phase. We do this by constructing an explicit mapping from any such code to a number of copies of Kitaev's code. Some of our results extend to some two-dimensional (2D) subsystem codes, including topological subsystem codes. Error correction benefits from the corresponding local mappings. In particular, it enables us to use decoding algorithm developed for Kitaev's code to decode any 2D stabilizer code and subsystem code.

We show how to construct a large class of quantum error-correcting codes, known as CalderbankSteane-Shor codes, from highly entangled cluster states. This becomes a primitive in a protocol that foliates a series of such cluster states into a much larger cluster state, implementing foliated quantum error correction. We exemplify this construction with several familiar quantum error-correction codes and propose a generic method for decoding foliated codes. We numerically evaluate the error-correction performance of a family of finite-rate Calderbank-Steane-Shor codes known as turbo codes, finding that they perform well over moderate depth foliations. Foliated codes have applications for quantum repeaters and fault-tolerant measurement-based quantum computation. DOI: 10.1103/PhysRevLett.117.070501 Quantum error correction is critical to building practical quantum-information processors (QIPs). In an influential series of papers, Raussendorf and co-workers described a measurement-based approach to fault-tolerant quantum processing using highly entangled cluster states, defined on a 3D lattice [1-4]. Raussendorf's 3D cluster state can be visualized as a foliation of Kitaev's surface code [5,6], i.e., a sequence of 2D surface-code "sheets," stacked together to form a 3D lattice. This is evident in Ref. [1], where it is shown that measuring the "bulk" qubits of a 3D cluster state leaves the two logical surface-code qubits encoded in the boundary faces in an entangled Bell pair.Raussendorf's 3D cluster gained prominence for its high fault-tolerant computational error thresholds ≲1%. It has applications in various QIP tasks, including long-range entanglement sharing, in which surface-code cluster states are created at regularly spaced local nodes, which are linked by medium-range optical channels into a 3D cluster state [7]. It is capable of fault-tolerant, measurement-based quantum computation, using an elegant geometric construction that braids defects in the interior of the 3D cluster state to produce robust Clifford gates. Universality is afforded by magic state injection and distillation [3,4,8].The robustness of Refs. [3,4] is inherited from the underlying surface code, which has a high error-correction threshold ∼11% [6,9-11]. The surface code has large distance and zero rate (in regards to the asymptotic ratio of the number of logical and physical qubits), reflecting the trade-off between distance and rate in two spatial dimensions [12]. It is natural to ask how to adapt the foliated structure of Refs. [1,2] to use other underlying codes that could achieve a higher encoding rate.Another motivation for our work is recent fault-tolerant schemes that produce a universal gate set by code deformation and code switching [13][14][15]. Extending code foliation to codes that circumvent magic state distillation [8] may produce cluster states with a lower resource overhead for fault-tolerant measurement-based QIPs.In this Letter we show that all Calderbank-Steane-Shor (CSS) codes can be clusterized, meaning that they can...

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We propose a simplified version of the Kitaev's surface code in which error correction requires only three-qubit parity measurements for Pauli operators XXX and ZZZ. The new code belongs to the class of subsystem stabilizer codes. It inherits many favorable properties of the standard surface code such as encoding of multiple logical qubits on a planar lattice with punctured holes, efficient decoding by either minimum-weight matching or renormalization group methods, and high error threshold. The new subsystem surface code (SSC) gives rise to an exactly solvable Hamiltonian with 3-qubit interactions, topologically ordered ground state, and a constant energy gap. We construct a local unitary transformation mapping the SSC Hamiltonian to the one of the ordinary surface code thus showing that the two Hamiltonians belong to the same topological class. We describe error correction protocols for the SSC and determine its error thresholds under several natural error models. In particular, we show that the SSC has error threshold approximately $0.6\%$ for the standard circuit-based error model studied in the literature. We also consider a model in which three-qubit parity operators can be measured directly. We show that the SSC has error threshold approximately 0.97% in this setting.

Topological quantum error-correcting codes are defined by geometrically local checks on a two-dimensional lattice of quantum bits (qubits), making them particularly well suited for fault-tolerant quantum information processing. Here, we present a decoding algorithm for topological codes that is faster than previously known algorithms and applies to a wider class of topological codes. Our algorithm makes use of two methods inspired from statistical physics: renormalization groups and mean-field approximations. First, the topological code is approximated by a concatenated block code that can be efficiently decoded. To improve this approximation, additional consistency conditions are imposed between the blocks, and are solved by a belief propagation algorithm.

We present a three-dimensional generalization of a renormalization group decoding algorithm for topological codes with Abelian anyonic excitations that we introduced for two dimensions in \cite{DP09a,DP10a}. We also provide a complete detailed description of the structure of the algorithm, which should be sufficient for anyone interested in implementing it. This 3D implementation extends our previous 2D algorithm by incorporating a failure probability of the syndrome measurements, i.e., it enables fault-tolerant decoding. We report a fault-tolerant storage threshold of $\sim1.9(4)\%$ for Kitaev's toric code subject to a 3D bit-flip channel (i.e. including imperfect syndrome measurements). This number is to be compared with the $2.9\%$ value obtained via perfect matching \cite{H04a}. The 3D generalization inherits many properties of the 2D algorithm, including a complexity linear in the space-time volume of the memory, which can be parallelized to logarithmic time.