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Hypergraph product (HGP) code[1–3]

Alternative names: Quantum hypergraph (QHG) code, Tillich-Zemor product code.

Description

A member of a family of CSS codes whose stabilizer generator matrix is obtained from a hypergraph product of two classical linear binary codes. Codes from hypergraph products in higher dimension are called higher-dimensional HGP codes [3].

More technically, the \(X\)- and \(Z\)-type stabilizer generator matrices of a hypergraph product code are, respectively, the boundary and coboundary operators of the 2-complex obtained from the tensor product of a chain complex and cochain complex corresponding to two classical linear binary seed codes. Let the two seed codes be \(C_i\) for \(i\in\{1,2\}\) with parameters \([n_i, k_i, d_i]\), defined as the kernel of \(r_i \times n_i\) check matrices \(H_i\) of rank \(n_i - k_i\). The hypergraph product yields two classical codes \(C_{X,Z}\) with parity-check matrices \begin{align} H_{X}&=\begin{pmatrix}H_{1}\otimes I_{n_{2}} & \,\,I_{r_{1}}\otimes H_{2}^{T}\end{pmatrix}\tag*{(1)}\\ H_{Z}&=\begin{pmatrix}I_{n_{1}}\otimes H_{2} & \,\,H_{1}^{T}\otimes I_{r_{2}}\end{pmatrix}~, \tag*{(2)}\end{align} where \(I_m\) is the \(m\)-dimensional identity matrix. These two codes then yield a hypergraph product code via the CSS construction.

In general, the stabilizer generator matrices of an \(m\)-dimensional hypergraph product code are the boundary and co-boundary operators of a 2-dimensional chain complex contained within an \(m\)-complex that is recursively constructed by taking the tensor product of an \((m-1)\)-complex and a 1-complex, with the 1-complex corresponding to some classical linear binary code.

Protection

If \([n_i, k_i, d_i]\) (\([r_i, k^T_i, d^T_i]\)) are the parameters of the codes \(\mathrm{ker}H_i\) (\(\mathrm{ker}H_i^T\), taking (\(d=\infty\) if \(k=0\)), the hypergraph product has parameters \([[n_1 n_2 + r_1 r_2, k_1 k_2 + k_1^T k_2^T, \min(d_1, d_2, d_1^T, d_2^T)]]\)

Encoding

Fault-tolerant state preparation via dimension jump [4].

Transversal Gates

Hadamard (up to logical SWAP gates) and control-\(Z\) on all logical qubits [5].Patch-transversal gates inherited from the automorphism group of the underlying classical codes [6; Appx. D].

Gates

Code deformation techniques yield Clifford gates [7].Targeted logical gates [8].Logical gates via Dehn twists for hypergraph products of cyclic codes [9].

Decoding

Single-ancilla syndrome extraction circuits do not admit hook errors [10].ReShape decoder that uses minimum weight decoders for the classical codes used in the hypergraph construction [11].2D geometrically local syndrome extraction circuits with depth order \(O(\sqrt{n})\) using order \(O(n)\) ancilla qubits [12].Improved BP-OSD decoder [13].Erasure correction can be implemented approximately with \(O(n^2)\) operations with quantum generalizations [14] of the peeling and pruned peeling decoders [15], with a probabilistic version running in \(O(n^{1.5})\) operations. Other nearly optimal erasure decoders exist [16,17]. Initial hypergraph product codes can be further optimized against the erasure channel using reinforcement learning [18].Syndrome measurements are distance-preserving because syndrome extraction circuits can be designed to avoid hook errors [19].Generalization [20] of Viderman's algorithm for expander codes [21].Linear time iterative decoder [22].

Fault Tolerance

Single-ancilla syndrome extraction circuits do not admit hook errors [10].There is a fault-tolerant universal computation scheme for hypergraph-product codes concatenated with the \([[4,2,2]]\) code [23].

Code Capacity Threshold

Some thresholds were determined in Ref. [24].Bounds on code capacity thresholds using ML decoding can be obtained by mapping the effect of noise on the code to a statistical mechanical model [25]. For example, a threshold of \(7\%\) was obtained under independent \(X\) and \(Z\) noise for codes obtained from random \((3,4)\)-regular Gallager codes.

Threshold

Circuit-level noise: \(0.1\%\) with all-to-all connected syndrome extraction circuits [12] and DiVincenzo-Aliferis syndrome extraction circuits [26] combined with non-local gates [27]. No threshold observed above physical noise rates at or above \(10^{-6}\) using 2D geometrically local syndrome extraction circuits.

Cousins

  • Locally testable code (LTC)— Applying the hypergraph product to an LTC yields a code which provides an explicit example of No Low-Error Trivial States (NLETS) [28].
  • XYZ product code— Hypergraph (XYZ) product codes are constructed out of hypergraph products of two (three) classical linear codes.
  • Linear binary code— Hypergraph product codes are constructed out of two linear binary codes.
  • Neural network quantum code— Using reinforcement learning, hypergraph product codes can be further optimized against the erasure channel [18] and can be weight reduced while maintaining distance [29].
  • \([[4,2,2]]\) Four-qubit code— There is a fault-tolerant universal computation scheme for hypergraph-product codes concatenated with the \([[4,2,2]]\) code [23].
  • Concatenated qubit code— There is a fault-tolerant universal computation scheme for hypergraph-product codes concatenated with the \([[4,2,2]]\) code [23].
  • Tiger surface code— The tiger surface code is constructed from a hypergraph product of two repetition codes over the integers.
  • Quantum locally recoverable code (QLRC)— A variant of the hypergraph product can be used to define QLRCs with intersecting recovery sets [30].
  • Single-shot code— Two-fold application of the hypergraph product to a pair of binary linear codes yields single-shot QLDPC codes that exploit redundancy in their stabilizer generators [31].
  • Self-correcting quantum code— There are bounds on the energy barrier of hypergraph product codes [32].
  • Quantum rainbow code— Hypergraph products of color codes yield quantum rainbow codes with growing distance and transversal gates in the Clifford hierarchy. In particular, utilizing this construction for quasi-hyperbolic color codes yields an \([[n,O(n),O(\log n)]]\) triorthogonal code family with magic-state yield parameter \(\gamma\to 0\) [33].
  • High-dimensional expander (HDX) code— Ramanujan codes utilize the hypergraph product with a twist, which is an automorphism on one of the complexes in the tensor product, in order to increase distance [34].
  • Rotated surface code— The rotated code can be obtained from hypergraph product of two cyclic linear binary codes with palindromic generator polynomial [2; Exam. 7].
  • Fractal surface code— The fractal product code is a hypergraph product of two classical codes defined on a Sierpinski carpet graph [35].
  • Subsystem homological product code— SP codes are projected higher-dimensional HGP codes [36].
  • Subsystem hypergraph product (SHP) code— Two SHP codes can be gauge-fixed to yield an HGP code [37; Sec. III]. The SHP and HGP code constructions yield the same dimension and minimum distance, but the former does not yield QLDPC codes; see [1; pg. 18].
  • Generalized bicycle (GB) code— An arbitrary qubit GB code of length \(2\ell\) is equivalent [38] to a rotated quantum hypergraph-product code with periodicity vectors \(\vec{L}_{1}\) and \(\vec{L}_{2}\) such that \(\lvert{\vec{L}_{1}\times\vec{L}_{2}\rvert=\ell}\).

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Coherent-parity-check (CPC) code
Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum
Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Fiber-bundle codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Homological product code
Parents
Homological product code
A homological product of chain complexes corresponding to two linear binary codes is a hypergraph product code [39]. Homological product codes constructed with diagonal boundary operators admit very different properties than those with rectangular boundary operators.
Quantum spatially coupled (SC-QLDPC) codeLattice stabilizer QLDPC Stabilizer Hamiltonian-based Qubit QECC Quantum
Hypergraph-product stabilizer generator matrices can be used as sub-matrices to define a 2D SC-QLDPC code [40].
Galois-qudit HGP codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Hypergraph product codes are Galois-qudit hypergraph-product codes for qudit dimension \(q=2\).
Hypergraph product (HGP) code
Children
Sierpinsky fractal spin-liquid (SFSL) code
The Sierpinsky fractal spin-liquid code is a hypergraph product of the repetition code and the Newman-Moore code [41,42].
Two-foliated fracton code
The two-foliated fracton code is a hypergraph product of the repetition code and the plaquette Ising code on a square lattice with periodic boundary conditions [43].
La-cross code
La-cross codes are constructed using a hypergraph product a cyclic LDPC code with itself.
Long-range enhanced surface code (LRESC)
LRESCs are constructed using a hypergraph product a concatenated LDPC-repetition code with itself.
Quantum expander code
Toric code
The toric code can be obtained from a hypergraph product of two repetition codes [2; Exam. 6]. Other hypergraph products of two repetition codes yield the related \([[2d^2-2d+1,1,d]]\) CSS code family [2; Exam. 5].

References

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[20]
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[21]
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[26]
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[37]
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[39]
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Zoo Code ID: hypergraph_product

Cite as:
“Hypergraph product (HGP) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/hypergraph_product
BibTeX:
@incollection{eczoo_hypergraph_product, title={Hypergraph product (HGP) code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hypergraph_product} }
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“Hypergraph product (HGP) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/hypergraph_product

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/qldpc/homological/balanced_product/hypergraph_product.yml.