Also known as 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].
Threshold
A lower bound on concatenated thresholds with CSS codes under biased noise [12].
Notes
See Ref. [13] for a brief review of asymmetric quantum codes.
Parent
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 [14].
- Clifford-deformed surface code (CDSC) — Random Clifford deformation can improve performance of surface codes agaisnt biased noise [15,16].
- XY surface code — XY surface codes perform well against biased noise [17].
- XYZ color code — XYZ color codes perform well against biased noise [18].
- Twisted XZZX toric code — Twisted XZZX codes perform well against biased noise [19–21]; see also Ref. [22].
- XZZX surface code — The XZZX surface code can be foliated for a noise-bias preserving MBQC [23] or FBQC [24] protocol; see also [25].
- Concatenated Steane code — Concatenating while taking into account noise bias can reduce resource overhead [11].
- 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 [26].
- Kitaev surface code — The surface code on a hexagonal lattice is an asymmetric CSS code [26].
- Compass code — Families of random compass codes perform well against biased noise and spatially dependent (i.e., asymmetric) noise [27].
- 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].
- Galois-qudit BCH code — Asymmetric quantum BCH codes have been constructed [2,28,29][4; Lemma 4.4], including subsystem BCH codes [6].
- Finite-geometry LDPC (FG-LDPC) code — EG-LDPC codes can be used to construct asymmetric CSS codes [4; Lemma 4.1].
- Galois-qudit RS code — Asymmetric Galois-qudit RS codes have been constructed [30,31].
- 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 [32]. A fault-tolerant teleportation-based computation scheme for asymmetric Bacon-Shor codes is effective against highly biased noise [33].
- Two-component cat code — Two-component cat codes admit a bias-preserving CNOT gate that is continuously connected to the identity [10,34].
- 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.
- GKP-surface code — Using rectangular lattices accounts for asymmetric noise and improves the GKP-surface threshold to \(0.58\) [35].
- Abelian LP code — The Abelian LP construction has been adapted to accommodate noise bias, yielding bias-tailored LP codes [36].
References
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Page edit log
- Victor V. Albert (2024-04-04) — most recent
Cite as:
“Asymmetric quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/asymmetric_qecc