Asymmetric quantum code

Asymmetric quantum code[1,2]

Alternative names: Noise-biased quantum code.

Description

Quantum systems can be roughly characterized by two types of noise, a bit-flip noise that maps canonical basis states into each other, and a phase-flip noise that induces relative phases between superpositions of such basis states. A code cannot protect against both types of noise arbitrarily well, and there is a tradeoff between the two types of protection. An asymmetric quantum code is one that performs much better against one type of noise than the other type. Such codes typically have tunable distances against each noise type and include CSS codes, GKP codes, and QSCs.

Protection

Noise channels for which one type of noise is more prominent than the other are called asymmetric-noise channels or biased-noise channels. An example of a noise-biased channel is a Pauli channel of independent \(X\) and \(Z\)-type noise with \(p_X \gg p_Z\) or visa versa.

In the context of comparing weight as well as of determining distances for noise models biased toward \(X\)- or \(Z\)-type errors, an extended notation for asymmetric CSS block quantum codes is \([[n,k,(d_X,d_Z)]]\) or \([[n,k,d_X/d_Z]]\), where \(d_{X,Z}\) are the \(X\)- and \(Z\)-distances, respectively [3]. An asymmetric Singleton bound and linear programming bounds for asymmetric CSS codes have been formulated [4], as well as asymmetric quantum GV bounds [5] and Hamming and Singleton bounds for general asymmetric subsystem codes [6]. Asymmetric MDS codes have been characterized [6,7].

Gates

Taking into account noise bias can reduce resource overhead in magic-state distillation schemes [8].A CNOT gate continuously connected to the identity cannot be noise-bias-preserving in finite dimensions [9][10; Appx. A].

Fault Tolerance

Fault-tolerant noise-bias-preserving computation scheme [9].Fault-tolerant circuits converting between asymmetric and symmetric subsystem codes [11,12].

Threshold

A lower bound on concatenated thresholds with CSS codes under biased noise [13].

Notes

See Ref. [14] for a brief review of asymmetric quantum codes.

Cousins

Distance-balanced code— Distance balancing is a procedure that can convert an asymmetric CSS code into a less asymmetric one.
Subsystem surface code— Subsystem surface codes perform well against biased circuit-level noise [15].
Clifford-deformed surface code (CDSC)— Random Clifford deformation can improve performance of surface codes against biased noise [16,17].
XY surface code— XY surface codes perform well against biased noise [18].
XYZ product code— XYZ product codes can be used to protect against biased noise [19].
XYZ color code— XYZ color codes perform well against biased noise [20].
Twisted XZZX toric code— Twisted XZZX codes perform well against biased noise [21–23]; see also Ref. [24].
XZZX surface code— The XZZX surface code can be foliated for a noise-bias preserving MBQC [25] or FBQC [26] protocol; see also [27].
Concatenated Steane code— Concatenating while taking into account noise bias can reduce resource overhead [12].
Quantum maximum-distance-separable (MDS) code— An asymmetric Singleton bound and linear programming bounds for asymmetric CSS codes have been formulated [4]. Asymmetric MDS codes have been characterized [7].
2D hyperbolic surface code— Asymmetric 2D hyperbolic surface codes have been constructed [28].
Kitaev surface code— The surface code on the honeycomb tiling is an asymmetric CSS code [28].
Calderbank-Shor-Steane (CSS) stabilizer code— In the context of comparing weight as well as of determining distances for noise models biased toward \(X\)- or \(Z\)-type errors, an extended notation for asymmetric CSS block quantum codes is \([[n,k,(d_X,d_Z),w]]\) or \([[n,k,d_X/d_Z,w]]\).
Quantum Reed-Muller code— Asymmetric quantum RM codes have been constructed [4; Lemma 4.1].
Finite-geometry LDPC (FG-LDPC) code— FG-LDPC codes can be used to construct asymmetric CSS codes [29][4; Lemma 4.1].
Hermitian code— Hermitian codes can be used to construct asymmetric Galois-qudit CSS codes [30].
Galois-qudit RS code— Asymmetric Galois-qudit RS codes have been constructed [31–33].
Bacon-Shor code— Bacon-Shor code parameters against bit- and phase-noise can be optimized by changing the block geometry, yielding good performance against biased noise [34]. A fault-tolerant teleportation-based computation scheme for asymmetric Bacon-Shor codes is effective against highly biased noise [35].
Two-component cat code— Two-component cat codes admit a bias-preserving CNOT gate that is continuously connected to the identity [10,36].
Square-lattice GKP code— GKP code parameters against position and momentum displacements can be tuned by the choice of lattice (e.g., square vs rectangular).
Binomial code— Binomial code parameters against loss/gain errors and dephasing can be tuned.
Quantum spherical code (QSC)— QSC code parameters against loss/gain errors and Gaussian rotations can be tuned.
Quantum parity code (QPC)— QPC parameters against bit- and phase-noise can be tuned.
EA qubit stabilizer code— Entanglement can help decode asymmetric quantum codes [37].
Hastings-Haah Floquet code— Floquet codes can be adapted for asymmetric noise [38].
GKP-surface code— Using rectangular lattices accounts for asymmetric noise and improves the GKP-surface threshold to \(0.58\) [39].
Tensor-network code— Quantum Lego and more general tensor-network code optimization for biased noise can be done using reinforcement learning [40,41].
Compass code— Families of random compass codes perform well against biased noise and spatially dependent (i.e., asymmetric) noise [42]. Clifford deformations can enhance the performance of compass codes against biased noise [43].
3D lattice stabilizer code— Applying Clifford deformations to various 3D stabilizer codes, including the 3D surface code, 3D color code, X-cube model code, the SFSL code, yields a \(50\%\) code capacity threshold under infinitely biased Pauli noise [17].
Galois-qudit BCH code— Asymmetric quantum BCH codes have been constructed [2,32,44,45][4; Lemma 4.4], including subsystem BCH codes [6,32].
Abelian LP code— The Abelian LP construction has been adapted to accommodate noise bias, yielding bias-tailored LP codes [46].

Member of code lists

Primary Hierarchy

Quantum Domain

Quantum code

Operator-algebra QECC (OAQECC)

Quantum error-correcting code (QECC)

Parents