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arXiv:2412.06744 · Quantum

Families of d=2 2D subsystem stabilizer codes for universal Hamiltonian quantum computation with two-body interactions

Phattharaporn Singkanipa, Zihan Xia, Daniel A. Lidar

Published Aug 5, 2025• 1 citation
arXivHTMLPDF

Abstract

In the absence of fault tolerant quantum error correction for analog, Hamiltonian quantum computation, error suppression via energy penalties is an effective alternative. We construct families of distance- 2 stabilizer subsystem codes we call “trapezoid codes'', that are tailored for energy-penalty schemes. We identify a family of codes achieving the maximum code rate, and by slightly relaxing this constraint, uncover a broader range of codes with enhanced physical locality, thus increasing their practical applicability. Additionally, we provide an algorithm to map the required qubit connectivity graph into graphs compatible with the locality constraints of quantum hardware. Finally, we provide a systematic framework to evaluate the performance of these codes in terms of code rate, physical locality, graph properties, and penalty gap, enabling an informed selection of error-suppression codes for specific quantum computing applications. We identify the [[4k+2,2k,g,2]] family of subsystem codes as optimal in terms of code rate and penalty gap scaling.

DOI
10.22331/q-2025-08-05-1821

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In the absence of fault tolerant quantum error correction for analog, Hamiltonian quantum computation, error suppression via energy penalties is an effective alternative. We construct families of distance- 2 stabilizer subsystem codes we call “trapezoid codes'', that are tailored for energy-penalty schemes. We identify a family of codes achieving the maximum code rate, and by slightly relaxing this constraint, uncover a broader range of codes with enhanced physical locality, thus increasing their practical applicability. Additionally, we provide an algorithm to map the required qubit connectivity graph into graphs compatible with the locality constraints of quantum hardware. Finally, we provide a systematic framework to evaluate the performance of these codes in terms of code rate, physical locality, graph properties, and penalty gap, enabling an informed selection of error-suppression codes for specific quantum computing applications. We identify the [[4k+2,2k,g,2]] family of subsystem codes as optimal in terms of code rate and penalty gap scaling.
We present general conditions for quantum error suppression for Hamiltonian-based quantum computation using subsystem codes. This involves encoding the Hamiltonian performing the computation using an error detecting subsystem code and the addition of a penalty term that commutes with the encoded Hamiltonian. The scheme is general and includes the stabilizer formalism of both subspace and subsystem codes as special cases. We derive performance bounds and show that complete error suppression results in the large penalty limit. To illustrate the power of subsystem-based error suppression, we introduce fully two-local constructions for protection against local errors of the swap gate of adiabatic gate teleportation and the Ising chain in a transverse field.
Quantum information is fragile and must be protected by a quantum error-correcting code for large-scale practical applications. Recently, highly efficient quantum codes have been discovered which require a high degree of spatial connectivity. This raises the question of how to realize these codes with minimal overhead under physical hardware connectivity constraints. Here, we introduce a general recipe to transform any quantum stabilizer code into a subsystem code that has local interactions, with weight and degree three, on a given graph. We call the subsystem codes produced by our recipe wire codes, and their code parameters depend on the input code and the given graph. Wire codes can be adapted to have a local implementation on any graph that supports a low-density embedding of the input Tanner graph, with an overhead that depends on the embedding. In particular, applying our results to a stabilizer code and a subdivision of its own Tanner graph, yields a quantum weight reduction procedure with a multiplicative qubit overhead and distance reduction that are linear in the input check degree and weight, respectively. Applying our results to hypercubic lattices leads to a construction of local subsystem codes with optimal scaling code parameters in any fixed spatial dimension. Similarly, applying our results to families of expanding graphs leads to local codes on these graphs with code parameters that depend on the degree of expansion. Our results constitute a general method to construct low-overhead subsystem codes on general graphs, which can be applied to adapt highly efficient quantum error correction procedures to hardware with restricted connectivity.
Quantum computers have been shown to efficiently solve a class of problems for which no efficient solution is otherwise known. Physical systems can implement quantum computation, but devising realistic schemes is an extremely challenging problem largely due to the effect of noise. A quantum computer that is capable of correctly solving problems more rapidly than modern digital computers requires some use of so-called fault-tolerant components. Code-based fault-tolerance using quantum error-correcting codes is one of the most promising and versatile of the known routes for fault-tolerant quantum computation. This dissertation presents three main, new results about code-based fault-tolerant quantum computer architectures. The first result is a large new family of quantum codes that go beyond stabilizer codes, the most well-studied family of quantum codes. Our new family of codeword stabilized codes contains all known codes with optimal parameters. Furthermore, we show how to systematically find, construct, and understand such codes as a pair of codes: an additive quantum code and a classical (nonlinear) code. Second, we resolve an open question about universality of so-called transversal gates acting on stabilizer codes. Such gates are universal for classical fault-tolerant computation, but they were conjectured to be insufficient for universal fault-tolerant quantum computation. We show that transversal gates have a restricted form and prove that some important families of them cannot be quantum universal. This is strong evidence that so-called quantum software is necessary to achieve universality, and, therefore, fault-tolerant quantum computer architecture is fundamentally different from classical computer architecture. Finally, we partition the fault-tolerant design problem into levels of a hierarchy of concatenated codes and present methods, compatible with rigorous threshold theorems, for numerically evaluating these codes. The methods are applied to measure inner error-correcting code performance, as a first step toward elucidation of an effective fault-tolerant quantum computer architecture that uses no more than a physical, inner, and outer level of coding. Of the inner codes, the Golay code gives the highest pseudothreshold of 2 × 10-3. A comparison of logical error rate and overhead shows that the Bacon-Shor codes are competitive with Knill's C 4/C6 scheme at a base error rate of 10 -4. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)
We present a new scheme to perform noise resilient universal adiabatic quantum computation using two-body interactions. To achieve this, we introduce a new family of error detecting subsystem codes whose gauge generators and a set of their logical operators - capable of encoding universal Hamiltonian computations - can be implemented using two-body interactions. Logical operators of the code are used to encode any given computational Hamiltonian, and the gauge operators are used to construct a penalty Hamiltonian whose ground subspace is protected against local-errors. In contrast to previous approaches, the constructed penalty Hamiltonian does not necessarily commute with the encoded computational Hamiltonians, but for our construction, the undesirable effect of the penalty Hamiltonian on the computation can be compensated by a simple modification of the implemented Hamiltonians. We also investigate whether a similar scheme can be constructed by encoding the computational Hamiltonian using only bare-logical operators of subsystem codes, to guarantee that computational Hamiltonian commutes with the penalty Hamiltonian. We prove a no-go theorem showing that restricting to two-body interactions and using any general CSS-type subsystem codes, such a construction cannot encode systems beyond an Ising chain in a transverse field. We point out that such a chain is universal for Hamiltonian-based quantum computation, but it is not universal for ground-state quantum computation.
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.
Orthogonal geometric constructions are the basis of many many quantum error-correcting codes (QEC), but strict orthogonality constraints limit design flexibility and resource efficiency. We introduce a quasi-orthogonal geometric framework for stabilizer codes that relaxes these constraints while preserving the symplectic commutation structure on the binary symplectic space F22\mathbb{F}_{2}^{2}F22​. The approach permits controlled overlap between X- and Z-check supports, leading to quasi-orthogonal Pauli operators and a generalized notion of effective distance defined via induced anti-commutation with logical operators. This relaxation expands the stabilizer design space, enabling codes that approach the Gilbert-Varshamov regime with improved logical rates at moderate distances. Finite-length constructions, including quasi-orthogonal variants of the [[8,3,≈3]][[8,3,\approx 3]][[8,3,≈3]], [[10,4,≈3]][[10,4,\approx 3]][[10,4,≈3]], [[13,1,5]][[13,1,5]][[13,1,5]], and [[29,1,11]][[29,1,11]][[29,1,11]] codes, demonstrate consistent improvements over strictly orthogonal counterparts. Under depolarizing noise with error rates up to p=0.30p=0.30p=0.30, logical error rates, fidelities, and trace distances improve by up to two orders of magnitude. These improvements reflect the increased connectivity of the underlying stabilizer geometry while remaining compatible with standard decoding schemes. The proposed framework offers a principled extension of stabilizer code design through quasi-orthogonal geometric structures.