Abhishek Rajput

Sequences of Bivariate Bicycle Codes from Covering Graphs

We show that given an instance of a bivariate bicycle (BB) code, it is possible to generate an infinite sequence of new BB codes using increasingly large covering graphs of the original code’s Tanner graph. When a BB code has a Tanner graph that is a $h$-fold covering of the base BB code’s Tanner graph, we refer to it as a $h$–$\textit{cover code}$. We show that for a BB code to be a $h$-cover code, its lattice parameters and defining polynomials must satisfy simple algebraic conditions relative to those of the base code. By extending the graph covering map to a chain map, we show there are induced projection and lifting maps on (co)homology that enable the projection and lifting of logical operators and, in certain cases, automorphisms between the base and the cover code. The search space of cover codes is considerably reduced compared to the full space of possible polynomials and we find that many interesting examples of BB codes, such as the $[[144,12,12]]$ gross code, can be viewed as cover codes. We also apply our method to search for BB codes with weight 8 checks and find many codes, including a $[[64,14,8]]$ and $[[144,14,14]]$ code. For an $h$-cover code of an $[[n,k,d]]$ BB code with parameters $[[n_h = hn, k_h \geq k, d \leq d_h \leq hd]]$, we prove that $k_h \geq k$ and $d_h \leq hd$ when $h$ is odd. Furthermore if $h$ is odd and $k_h = k$, we prove the lower bound $d \leq d_h$. We conjecture it is always true that an $h$-cover BB code of a base $[[n,k,d]]$ BB code has parameters $[[n_h = hn, k_h \geq k, d \leq d_h \leq hd]]$. While the focus of this work is on bivariate bicycle codes, we expect these methods to generalise readily to many group algebra codes. 

Alexander Cowtan

The Spaces Between: performing quantum code surgery without idling

In reducing the spacetime overhead of surgery with quantum LDPC codes, there has been a focus on increasing parallelism and reducing the number of syndrome rounds per surgery operation (“single-shot surgery”). An oft-overlooked overhead is the idling time between rounds of surgery, present to prevent low-weight logical faults which extend between rounds. Generic upper bounds imply that O(d) rounds are sufficient, but this would be a burdensome slowdown on a real quantum computer, and is widely believed to be unnecessary in practice.

I will first describe a formalism which captures the spacetime volume of surgery operations performed sequentially, by taking mapping cones on 4-term fault complexes. I will then give results which prove when idling is unnecessary and waiting only O(1) rounds is sufficient. These results come in two flavours: (A) conditions on the properties of the ancilla systems used for surgery, and (B) conditions on the local-testability of the original code.

Anqi Gong

Logical gates in some algebraic codes

Algebraic structures in quantum error-correcting codes can usually facilitate logical computation. Polynomial formalism provides a way to investigate the Clifford and non-Clifford gate possibilities in these codes. I will give two types of examples in this talk. The first is Reed-Muller codes; both the punctured versions encoding one logical qubit and a high-rate family but with prudent gauge-fixing will be presented. The latter enables one to implement arbitrary in-block logical CNOT gates through physical permutations only, thereby having the potential to speed up logical computation. The second type concerns small algebraic geometry codes. Based on these codes, we give various practically relevant magic state distillation protocols and propose their application in certain algorithm subroutines.

Basudha Srivastava

Sequential decoding of concatenated codes

Christopher Pattison

Fault-tolerant quantum computer with a constant number of control lines

Delivering control signals to qubits is a substantial bottleneck for many quantum computing platforms. In this work, we give a construction of a fault-tolerant 2D geometrically-local quantum processor where computation is implemented by the repeated application of a hard-coded constant-depth gate sequence along with individual control of a constant number of qubits analogous to I/O pins in classical integrated circuits. For platforms where global driving of qubits or local fixed control pulse generation is possible, we interpret our construction as a 2D quantum processor with unit density of wires and a constant number of wires to the outside world: The repeated circuit may be implemented by broadcasting gate signals to large numbers of qubits analogous to clock signals distributed by clock distribution networks in digital integrated circuits. While the construction is unlikely to be practical in the near term, the scheme substantially benefits from the tight integration of classical digital logic with high density qubit technologies and may become feasible after further technology development.

Dongjin Lee

Chiral Color Code: Single-shot error correction for exotic topological order and beyond.

In this talk, I will introduce a family of simple three-dimensional stabilizer codes, called the chiral color codes, that realize fermionic and chiral topological orders. In the qubit case, the code realizes the topological phase of a single copy of the fermionic toric code. For qudit systems with local dimension $d$ the model features a chiral parameter $\alpha$ and realizes 3D topological phases characterized by $Z_d^\alpha$ anyon theories with anomalous chiral surface topological order. Furthermore, we prove that the bulk is short-range entangled (for odd $d$, coprime $\alpha$) by constructing an explicit local quantum channel that prepares the ground state. The chiral color codes are constructed within the gauge color code, and hence inherit its fault-tolerant features: they admit single-shot error correction and allow code switching to other stabilizer color codes. These properties position the chiral color codes as particularly useful platforms for realizing and manipulating fermions and chiral anyons. In the last part of the talk, I will also explain chiral stabilizer mixed states that arise at the boundary of chiral color codes, which cannot be diagnosed by the either modular commutator or chiral central charge.

Emma Rosenfeld

Magic state cultivation on a superconducting quantum processor

Fault-tolerant quantum computing requires a universal gate set, but the necessary non-Clifford gates represent a significant resource cost for most quantum error correction architectures. Magic state cultivation offers an efficient alternative to resource-intensive distillation protocols; however, testing the proposal’s assumptions represents a challenging departure from quantum memory experiments. We present an experimental study of magic state cultivation on a superconducting quantum processor. We implement cultivation, including code-switching into a surface code, and develop a fault-tolerant measurement protocol to bound the magic state fidelity. Cultivation reduces the error by a factor of 40, with a state fidelity of 0.9999(1) (retaining 8% of attempts). Our results experimentally establish magic state cultivation as a viable solution to one of quantum computing’s most significant challenges.

Esha Swaroop

Universal Adapters between Quantum LDPC codes

In the last few years, quantum LDPC codes have developed to become serious contenders for the fault-tolerant architecture needed to build a large-scale quantum computer. This interest has spurred recent developments in low-overhead computing with these codes, such as by leveraging inherent symmetries present in the quantum LDPC code, or by deforming arbitrary codes. In this talk, we will focus on the latter approach, and discuss extensions to quantum LDPC surgery following the work of Hastings [arXiv:2102.10030] and Cohen et al. [10.1126/sciadv.abn1717, 2022].

Our main result is the measurement of joint logical Pauli operators between arbitrary quantum LDPC codes using ancillary qubits hosting a repetition code of equivalent distance, which functions as a code adapter for logical gates as well as code switching. This adapter is universal in the sense that it works regardless of the LDPC codes involved and the logical Paulis being measured. The number of additional qubits scales quasi-linearly, or in some cases linearly, with the weight of the Pauli logical operator to be measured.

Using a similarly constructed adapter, we also show a way to unitarily implement targeted logical gates on arbitrary LDPC codes, by efficiently mediating with a code hosting the required symmetries for more exotic gates.

Friederike Butt

Code switching for measurement-free quantum computing

The ability to perform quantum error correction (QEC) and robust gate operations on encoded qubits is a key step toward practical demonstrations of quantum algorithms. Contemporary QEC schemes typically require mid-circuit measurements with feed-forward control, which are challenging for qubit control, often slow, and susceptible to relatively high error rates. I present protocols for a measurement-free universal toolbox of fault-tolerant logical operations based on code switching between two- and three-dimensional colour codes. Individually, neither two- nor three-dimensional topologies support a universal transversal gate set {H, T, CNOT}, with the T-gate missing in the two-dimensional and the H-gate in the three-dimensional case. However, by transferring encoded information between a two- and a three-dimensional code while preserving the logical state gives access to a full transversal universal gate set. I discuss the conditions required for fault-tolerant code switching and how these conditions can be satisfied in practice. Building on this, I extend code switching to a fully measurement-free setting, where protocols are implemented using coherent fault-tolerant circuits that avoid feed-forward during algorithm execution. Finally, I demonstrate that this measurement-free toolbox enables the execution of fault-tolerant quantum algorithms, exemplified by a realization of Grover’s search on encoded logical qubits.

Grace Sommers

Spectral properties and coding transitions of Haar-random quantum codes

Abstract: A quantum error-correcting code with a nonzero error threshold undergoes a mixed-state phase transition when the error rate reaches that threshold. We explore this phase transition for Haar-random quantum codes, in which the logical information is encoded in a random subspace of the physical Hilbert space. We focus on the spectrum of the encoded system density matrix as a function of the rate of uncorrelated, single-qudit errors. For low error rates, this spectrum consists of well-separated bands, corresponding to errors of different weights. As the error rate increases, the bands for high-weight errors merge. The evolution of these bands with increasing error rate is well described by a simple analytic ansatz. Using this ansatz, as well as an explicit calculation, we show that the threshold for Haar-random quantum codes saturates the hashing bound, and thus coincides with that for random stabilizer codes. For error rates that exceed the hashing bound, typical errors are uncorrectable, but postselected error correction remains possible until a much higher detection threshold. Postselection can in principle be implemented by projecting onto subspaces corresponding to low-weight errors, which remain correctable past the hashing bound.

Josias Old

Addressable fault-tolerant universal quantum gate operations for high-rate lift-connected surface codes

Quantum low-density parity check (qLDPC) codes are among the leading candidates to realize error-corrected quantum memories with low qubit overhead. Potentially high encoding rates and large distance relative to their block size make them appealing for practical suppression of noise in near-term quantum computers. In addition to increased qubit-connectivity requirements compared to more conventional topological quantum error correcting codes, qLDPC codes remain notoriously hard to compute with. In this work, we introduce a construction to implement all Clifford quantum gate operations on the recently introduced lift-connected surface (LCS) codes (Old et al. 2024). These codes can be implemented in a 3D-local architecture and achieve asymptotic scaling [[n, O(n^1/3), O(n^1/3)]] . In particular, LCS codes realize favorable instances with small numbers of qubits: For the [[15,3,3]]-LCS code, we provide deterministic fault-tolerant (FT) circuits of the logical gate set {H_i, S_i, C_iX_j}_{i,j in (0,1,2)} based on flag qubits. By adding a procedure for FT magic state preparation, we show quantitatively how to realize an FT universal gate set in d=3 LCS codes. Numerical simulations indicate that our gate constructions can attain pseudothresholds in the range of 4.8*10^-3 – 1.2*10^-2 for circuit-level noise. The schemes use a moderate number of qubits and are therefore feasible for near-term experiments, facilitating progress for fault-tolerant error corrected logic in high-rate qLPDC codes.

Dynamic Compass Code

Michael Vasmer

Fault-tolerant transformations of spacetime codes

Recent advances in quantum error-correction (QEC) have shown that it is often beneficial to understand fault-tolerance as a dynamical process, a circuit with redundant measurements that help correct errors, rather than as a static code equipped with a syndrome extraction circuit. Spacetime codes have emerged as a natural framework to understand error correction at the circuit level while leveraging the traditional QEC toolbox. Here, we introduce a framework based on chain complexes and chain maps to model spacetime codes and transformations between them. We show that stabilizer codes, quantum circuits, and decoding problems can all be described using chain complexes, and that the equivalence of two spacetime codes can be characterized by specific maps between chain complexes, the fault-tolerant maps, that preserve the number of encoded qubits, fault distance, and minimum-weight decoding problem. As an application of this framework, we extend the foliated cluster state construction from stabilizer codes to any spacetime code, showing that any Clifford circuit can be transformed into a measurement-based protocol with the same fault-tolerant properties. To this protocol, we associate a chain complex which encodes the underlying decoding problem, generalizing previous cluster state complex constructions. Our method enables the construction of cluster states from non-CSS, subsystem, and Floquet codes, as well as from logical Clifford operations on a given code.

Joint work with Arthur Pesah, Austin K. Daniel, and Ilan Tzitrin https://arxiv.org/abs/2509.09603

Paul Webster