Session N64: Quantum Error Correction Foundations and Theory I

Entanglement Purification with Quantum LDPC Codes and Iterative Decoding

Recent constructions of quantum LDPC (QLDPC) codes provide optimal scaling of the code parameters, enabling fault-tolerant quantum systems with minimal resource overhead. However, the hardware path to long-range-interaction-demanding QLDPC codes is likely a challenging one. Given the practical difficulty in building a monolithic architecture for quantum systems, such as computers, based on optimal QLDPC codes, it is worth considering a distributed implementation of such codes over a network of interconnected medium-sized quantum processors. In such a setting, all syndrome measurements and logical operations must be performed through the use of high-fidelity shared entangled states between the processing nodes. Since probabilistic many-to-1 distillation schemes for purifying entanglement are inefficient, we investigate quantum error correction based entanglement purification in this work. Specifically, we employ QLDPC codes to distill GHZ states, as the resulting high-fidelity logical GHZ states can interact directly with the code used to perform distributed quantum computing (DQC), e.g. for fault-tolerant Steane syndrome extraction. We use the min-sum decoder for distilling 3-qubit GHZ states using a rate 0.118 family of lifted product QLDPC codes and obtain a threshold of 10.7% under depolarizing noise. Our results can be extended to larger size GHZ states as well.   

Quantum two-block non-abelian group codes

We study numerically and analytically a class of quantum two-block CSS codes with generator matrices H_X = (A,B), H_Z = (B^T,A^T), where commuting square matrices A, B of size m × m are associated with elements of the group algebra F₂G, and G is a group of order m. These codes are the shortest lifted-product codes [1] that can be obtained from the group G. Also, they contain all generalized-bicycle (GB) codes obtained when G is the (abelian) group of cyclic permutations. The two-block group code (TBGC) ansatz is of interest as a method to construct short quantum LDPC codes without the upper distance bound of more regular GB codes [2]. Analytically, we show that non-trivial TBGCs with stabilizer generators of weight w = 4 can be always expressed as direct sums of GB codes. We also compute optimal parameters of TBGCs associated with non-abelian groups of orders m ≤ 50 by exhaustive enumeration of all such codes.

[1] P. Panteleev and G. Kalachev, “Asymptotically good quantum and locally testable classical LDPC codes,” arXiv:2111.03654

[2] R. Wang and L. P. Pryadko “Distance bounds for generalized bicycle codes,” arXiv:2203.17216   

Constant-Overhead Quantum Error Correction with Thin Planar Connectivity

The surface code is the most popular quantum error correction code for the design of large-scale fault-tolerant quantum computers. However, it leads to a huge overhead because each logical qubit is encoded into hundreds of physical qubits. In this talk, I will discuss our proposal for a low overhead quantum error correction scheme based on quantum LDPC codes and a few layers of long-range connections.

    

Spacetime topological defect networks and floquet codes

I propose an extension of the topological defect network construction from groundstate wave functions to spacetime partition functions as a unifying framework to describe static fracton topological orders and dynamical topological floquet codes.   

Qutrit-based topological subsystem codes

We introduce a topological subsystem code based on a generalization of Kitaev's honeycomb model to qutrits. The code exhibits a number of beneficial properties, including a realization on a low-degree graph in a planar geometry, two-body checks, and high biased-noise error thresholds. We further demonstrate that topological twist defects can be constructed by two-body measurements, and the full Clifford group can be implemented fault tolerantly by braiding twist defects. Through a mapping to a statistical mechanical model, we compute the optimal error thresholds of the subsystem code. We also establish a mapping of the error thresholds to those of a Z₃ surface code with anisotropic pure Pauli X noise.   

Pauli topological subsystem codes from Abelian anyon theories

We construct Pauli topological subsystem codes characterized by arbitrary two-dimensional Abelian anyon theories -- this includes non-modular anyon theories and those without a Lagrangian subgroup. We exemplify the construction with topological subsystem codes based on the non-modular Z₄⁽¹⁾ anyon theory and the chiral semion theory, both of which cannot be captured by topological stabilizer codes. The construction proceeds by "gauging out" certain anyon types of a topological stabilizer code. This amounts to defining a gauge group generated by the stabilizer group of the topological stabilizer code and a set of anyonic string operators for the anyon types that are gauged out. The resulting topological subsystem code is characterized by an anyon theory that is a proper subset of the anyons of the topological stabilizer code. We thereby show that every Abelian anyon theory is a subtheory of a stack of toric codes and a certain family of twisted quantum doubles that generalize the double semion anyon theory. Our work thus extends the classification of two-dimensional Pauli topological subsystem codes to systems of composite-dimensional qudits and shows that the classification is at least as rich as that of two-dimensional Abelian anyon theories. We further prove a number of general statements about the logical operators of translation invariant topological subsystem codes and define their associated anyon theories in terms of higher-form symmetries.   

Quantum error correction in a time-dependent transverse field Ising model

We describe a simple quantum error correcting code built out of a time-dependent transverse field Ising model. The code is similar to a repetition code, but has two advantages: an N-qubit code can be implemented with a finite-depth spatially local unitary circuit, and it can subsequently protect against both X and Z errors if N≥10 is even. We propose an implementation of this code with 10 ultracold Rydberg atoms in optical tweezers, along with further generalizations of the code.   

Homomorphic Logical Measurements

Shor and Steane ancilla are two well-known methods for fault-tolerant logical measurements,

which are successful on small codes and their concatenations. On large quantum low-density-parity-

check (LDPC) codes, however, Shor and Steane measurements have impractical time and space over-

head respectively. In this work, we widen the choice of ancilla codes by unifying Shor and Steane

measurements into a single framework, called homormophic measurements. As an example, we utilize

the theory of covering spaces to construct homomorphic measurement protocols for arbitrary X- or

Z-type logical Pauli measurements on surface codes in general, including the toric code and hyper-

bolic surface codes. These protocols do not require repetitive measurements or complicated ancilla

state preparation procedures such as distillation, which overcomes the difficulties of both Shor and

Steane methods. In contrast to other existing methods such as lattice surgery, conventional surface

code decoders can be directly applied to our constructions. The conceptual idea of our framework can

be extended to other quantum LDPC codes, which presents a new direction of fault-tolerant logical

operation design.   

Hyperbolic Floquet Quantum Error Correcting Codes

Large-scale universal quantum computation requires protecting quantum information against quantum noise which can be performed using quantum error-correcting codes. Recently, Hastings and Haah [1] introduced a fault-tolerant dynamical subsystem code on a honeycomb lattice using two-qubit check measurements. Despite having zero logical qubits when viewed as a static subsystem code, the sequence of two-qubit measurements protects two dynamical logical qubits. On the other hand, hyperbolic surface codes are known as a generalization of toric code on a negatively curved manifold. Since the genus of such manifolds increases with the system size, it has been demonstrated that these codes possess a constant encoding rate. In this work, we introduce hyperbolic Floquet codes, where two-qubit check operators are measured in a periodic sequence. Using this measurement scheme, first, we show that the number of dynamically protected logical qubits grows with the system size. Next, we numerically demonstrate the existence of a threshold for the error rate of physical qubits that our code can protect against. Finally, we propose a planar version of these codes with open boundary conditions which are experimentally more feasible to be realized using current platforms.

[1] Hastings, M. B., and Jeongwan H., "Dynamically generated logical qubits" Quantum 5 (2021): 564   

Quantum Perspectives on the Clifford Group

The Clifford group, a multiplicative group generated by the Hadamard, phase, and CNOT gates, defines a set of unitary operations which normalize the Pauli group. The finite structure of this group promotes a graph-theoretic description, known as a Cayley graph, with vertices indicating each group element and edges representing the particular generators that transform Clifford operations into each other. Acting with the Clifford group on a computational basis state generates the complete set of stabilizer states, the set of all n-qubit quantum states invariant under some 2ⁿ-element subset of the n-qubit Pauli group. Stabilizer states and stabilizer circuits, those quantum circuits exclusively composed of Clifford operations and stabilizer measurements, are famously known to be simulable on a classical computer. The Hilbert space of n-qubit stabilizer states also admits a natural description as a mathematical graph, known as a reachability graph, which can be constructed as a quotient space of the overall Cayley graph after identifying group elements which act trivially on each stabilizer state. When considering the action of some Clifford subgroup this quotient space separates into multiple disconnected subgraphs, which we term restricted graphs. We introduce this definition of reachability graphs as Cayley graph quotient spaces and demonstrate how operator identities on stabilizer states completely describe the identifications that produce restricted graphs. Motivated by an understanding of stabilizer state entropies, we provide a complete description of the Clifford subgroup generated by the Hadamard and CNOT gates acting on any pair of qubits in an n-qubit system, and present some physical insights gained from studying the associated set of restricted graphs.   

Multi spin Clifford code for higher order angular momentum errors in spin systems

The physical symmetries of a system play a central role in quantum error correction. In this work we encode a qubit in a collection of systems with angular-momentum symmetry (spins), extending the tools developed in [1] for single large spins. By considering large spins present in atomic systems and focusing on their collective symmetric subspace, we develop new codes with octahedral symmetry capable of correcting errors up to second order in angular-momentum operators. These errors include the most physically relevant noise sources such as microwave control errors and optical pumping. We additionally explore new qubit codes that exhibit distance scaling commensurate with the surface code while permitting transversal single-qubit Clifford operations.

[1] Phys. Rev. Lett. 127, 010504   

Preparing the XY surface code with high threshold under biased noise

Tailoring quantum surface codes by local Clifford deformation can increase their thresholds under biased noise. A specific example is the XY-surface code, which reduces to a repetition code under pure dephasing noise and achieves code capacity threshold of 50%. However, it is crucial to analyze the performance of the code during logical operations such as state preparation. In the standard approach, a logical X (or Y) state is prepared by initializing each physical qubit in the |+> (or |+i>) state, followed by measuring all stabilizers. However, this technique breaks the underlying symmetry of the XY code under pure dephasing noise, which limits the logical state preparation threshold. In this work, we propose a new logical initialization protocol which maintains the effectiveness of the XY code against dephasing noise. In this protocol, physical qubits are first locally entangled into two- or four-body Bell states, following which all the stabilizers are measured. We prove that in this procedure dephasing errors can be decoded as a repetition code, which guarantees a 50% state preparation threshold. Our analysis is supported by numerical simulations, which also show an overall improvement in threshold when dephasing errors is accompanied by small amount of bit-flip noise.