Quantum repetition code

Quantum repetition code[1]

Description

Encodes \(1\) qubit into \(n\) qubits according to \(|0\rangle\to|\phi_0\rangle^{\otimes n}\) and \(|1\rangle\to|\phi_1\rangle^{\otimes n}\). The code is called a bit-flip code when \(|\phi_i\rangle = |i\rangle\), and a phase-flip code when \(|\phi_0\rangle = |+\rangle\) and \(|\phi_1\rangle = |-\rangle\).

The \(\pm\)-basis codewords for the bit-flip code are GHZ states [2–4] (a.k.a. qubit cat states) \(|0\rangle^{\otimes n}\pm|1\rangle^{\otimes n}\). These are ground states of the one-dimensional classical Ising model Hamiltonian \(H=\sum_{i} Z_{i}Z_{i+1}\).

The \(\pm\)-basis codewords for the phase-flip code are expanded in the computational basis as \begin{align} \begin{split} |\overline{+}\rangle =\frac{1}{\sqrt{2^{n-1}}}\sum_{\sum_{i}v_{i}=0}|v_{1},\cdots,v_{n}\rangle~{\phantom{,}}\\ |\overline{-}\rangle =\frac{1}{\sqrt{2^{n-1}}}\sum_{\sum_{i}v_{i}=1}|v_{1},\cdots,v_{n}\rangle~, \end{split} \tag*{(1)}\end{align} showing that the phase-flip code stores information in the total parity of the qubits. For example, an early code realized in devices is the 2-qubit phase-flip code [5], which encodes a logical qubit into Bell states \(|00\rangle+|11\rangle\) and \(|01\rangle+|10\rangle\).

Protection

Bit-flip code detects bit-flip errors \(X\) on \(\left\lfloor (n-1)/2\right\rfloor\) qubits and does not detect any phase-flip errors \(Z\). Phase-flip code detects phase-flip errors \(Z\) on \(\left\lfloor (n-1)/2\right\rfloor\) qubits and does not detect any bit-flip errors \(X\).

Because they protect against only one type of noise, both codes can be thought of as a classical \([n,1,d]\) repetition code with classical distance \(d=\left\lfloor (n-1)/2\right\rfloor\) embedded in a quantum system. Nevertheless, the phase-flip code can offer some degree of protection in particular physical systems based on superconducting circuits [6,7].

Encoding

Non-deterministic encoders for various specific states of the 2-qubit phase-flip code [8].Fault-tolerant GHZ-state preparation with small qubit registers [9].Unitary circuit of depth logarithmic in \(n\) [10]. Any circuit has to have range \(n\) because Ghz states are locally indistinguishable [11].Adaptive constant-depth circuit with geometrically local gates and measurements throughout [12,13].Lindbladian-based dissipative encoding and autonomous QEC passively protecting against bit flips [14,15].Approximate Hamiltonian-based encoding of GHZ states [16].

Gates

Toffoli magic-state preparation protocol [17].

Decoding

Fault-tolerant syndrome detection [18].Autonomous QEC for the 3-qubit bit-flip code [19].Machine learning algorithm to implement autonomous QEC for the three-qubit quantum repetition code [20].Quantum version of the Tsirelson local automaton decoder [21].Planar decoder designed to work under circuit-level noise [22].

Fault Tolerance

Fault-tolerant syndrome detection [18].Toffoli magic-state preparation protocol [17].

Code Capacity Threshold

Independent \(X\) noise: \(50\%\) with RG decoder for quantum repetition code arranged on a 1D or 2D lattice [23].

Threshold

Phenomenological noise: \(11\%\) and \(17.2\%\) with RG decoder for quantum repetition code arranged on a 1D and 2D lattice, respectively [23].

Realizations

NMR: 2-qubit phase-flip code [24,25]; 3-qubit bit-flip code [26]; 3-qubit phase-flip code [27,28], with up to two rounds of error correction in liquid-state NMR [29]. Such codes were used to characterize noise [30].Linear optics: 2-qubit phase-flip code [31].Trapped ions: 3-qubit bit-flip code by Wineland group [32], and 3-qubit phase-flip algorithm implemented in 3 cycles on high fidelity gate operations [33]. Both phase- and bit-flip codes for 31 qubits and their stabilizer measurements have been realized by Quantinuum [34]. Multiple rounds of Steane error correction [35].Superconducting circuits: 3-qubit phase-flip and bit-flip code by Schoelkopf group [36,37]; 3-qubit bit-flip code [38]; 3-qubit phase-flip code up to 3 cycles of error correction [39]; IBM 15-qubit device [40]; IBM Rochester device using 43-qubit code [41]; Google system performing up to 8 error-correction cycles on 5 and 9 qubits [42]; Google Quantum AI Sycamore utilizing up to 11 physical qubits and running 50 correction rounds [43]; Google Quantum AI Sycamore utilizing up to 25 qubits for comparison of logical error scaling with a quantum code [44] (see also [45]). Google Quantum AI follow-up experiment on codes up to (classical) distance 29, demonstrating exponential suppression to an error floor of \(10^{-10}\) [46]. Ising-model Nishimori phase transition realized for GHZ states on 54 qubits on a 127 qubit IBM device [47]. GHZ state on 75 qubits made on an IBM device [48]. Implementation of planar decoder for codes with distances between 3 and 11 on 72-qubit superconducting device [22]. Lattice surgery on the surface-17 code has been realized by splitting the code into two repetition codes by the Wallraff group [49]. Autonomous QEC protocols have been implemented on a 3-qubit superconducting qubit device [50].Semiconductor spin-qubit devices: 3-qubit devices at RIKEN [51] and Delft [52].Nitrogen-vacancy centers in diamond: 3-qubit phase-flip code [53,54] (see also Ref. [55]).Repetition codes are used in quantum annealing protocols [56–58].

Notes

Repetition codes can be used to benchmark device performance [59].

Cousins

Fracton stabilizer code— The 1D quantum repetition code is an ingredient in product constructions that yield several fracton phases [60; Fig. 8].
Abelian topological code— The 1D quantum repetition code is an ingredient in product constructions that yield several topological phases [60; Fig. 8].
Cluster-state code— GHZ states can be used as resource states for MBQC protocols [61–63].
Perfect-tensor code— GHZ states are \(1\)-uniform for all \(n\) and AME for \(n=2,3\).
Repetition code— A quantum repetition code can be thought of as a classical \([n,1,d]\) repetition code with classical distance \(d=\left\lfloor (n-1)/2\right\rfloor\) embedded in a quantum system.
Very small logical qubit (VSLQ) code— Parts of the VSLQ codewords resemble the two-qubit phase-flip repetition code, though the code cannot correct phase errors. Unlike the phase-flip code, the VSLQ code can correct for single photon loss because it uses the second excited state in the construction, which remains distinct from the vacuum even after photon loss.
Cat-repetition code— Two-component cat codes in the cat-state basis have been concatenated with quantum repetition codes [17,64–67].
Two-component cat code— Two-legged cat and quantum repetition codes can be thought of as classical codes because they protect against only one type of noise. Two-legged cat codes (quantum repetition) codes suppress cavity dephasing (bit-flip) noise exponentially with \(|\alpha|^2\) (\(n\)). The stability offered by cat codes has been linked to several favorable properties of phases of matter associated with the repetition-code Hamiltonian [68,69].
\(D_4\) hyper-diamond GKP code— The \(D_4\) hyper-diamond GKP code can be seen as a concatenation of a rotated square-lattice GKP code with a repetition code [70]. This is related to the fact that the four-bit repetition code yields the \(D_4\) hyper-diamond lattice via Construction A.
Concatenated GKP code— Concatenating a three-qubit quantum repetition code with GKP codes can correct some two-bit-flip errors [71] (see also [72]).
Coherent-state repetition code— Two-component cat codes in the coherent-state basis have been concatenated with quantum repetition codes [73,74].
Self-correcting quantum code— The bit-flip repetition code associated with the 2D classical Ising model is a self-correcting classical memory [75; Sec. V.A].
XYZ ruby Floquet code— One third of the time during its measurement schedule, the ISG of the XYZ ruby Floquet code is that of the 6.6.6 color code concatenated with a three-qubit repetition code.
Kitaev chain code— The Kitaev chain code can be thought of as the Majorana stabilizer analogue of the quantum repetition code [76].
\([[2^D,D,2]]\) hypercube quantum code— The hypercube quantum code can be concatenated with a two-qubit quantum repetition code to yield a \([[2^{D+1},D,4]]\) error-detecting code family [77].
\([[9,1,3]]\) Shor code— The Shor code is a concatenation of a three-qubit bit-flip with a three-qubit phase-flip repetition code.
Transverse-field Ising model (TFIM) code— When written in the computational basis, the phase-flip and TFIM codewords are superpositions of qubit states of fixed total parity. The superposition is equal for the phase-flip code, whereas some states appear with a \(-1\) coefficient for TFIM code. However, the TFIM code can be encoded in constant depth.
X-cube model code— Generalized X-cube models [60] are constructed from a balanced product of the quantum repetion (1D Ising) code and the Newman-Moore code.
Modular-qudit shift-resistant code— Both the quantum repetition and modular-qudit shift-resistant codes protect against only one type of noise.

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 Shor code

Quantum parity code (QPC)Concatenated quantum QECC Quantum

Parents

Quantum parity code (QPC)

A \([[m_1 m_2,1,\min(m_1,m_2)]]\) QPC reduces to a repetition code when \(m_1\) or \(m_2\) is one.

Group-based quantum repetition codeConcatenated quantum QECC Quantum

Group-based quantum repetition codes reduce to quantum repetition codes for \(G = \mathbb{Z}_2\).

GNU PI codeQubit PI Cyclic quantum Amplitude-damping (AD) QECC AQECC Quantum

GNU codewords for \(g=1\) reduce to the phase-flip repetition code.

Frustration-free Hamiltonian codeHamiltonian-based QECC Quantum

The codespace of the quantum repetition code is the ground-state space of a frustration-free classical Ising model with nearest-neighbor interactions.

Commuting-projector Hamiltonian codeHamiltonian-based QECC Quantum