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 [2,3] (a.k.a. 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.

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 [4,5].

Encoding

Unitary circuit of depth logarithmic in \(n\) [6]. Any circuit has to have range \(n\) because Ghz states are locally indistinguishable [7].Adaptive constant-depth circuit with geometrically local gates and measurements throughout [8,9].Lindbladian-based dissipative encoding passively protecting against bit flips [10,11].

Gates

Toffoli magic-state preparation protocol [12].

Decoding

Automaton-like decoders for the repetition code on a 2D lattice, otherwise known as the classical 2D Ising model, were developed by Toom [13,14]. An automaton by Gacs [15,16] and Tsirelson [17] yield local decoders for a 1D lattice.Machine learning algorithm to implement continuous error-correction for the three-qubit quantum repetition code [18].

Fault Tolerance

Toffoli magic-state preparation protocol [12].

Code Capacity Threshold

Independent \(X\) noise: \(50\%\) with RG decoder for quantum repetition code arranged on a 1D or 2D lattice [19].

Threshold

Phenomenological noise: \(11\%\) and \(17.2\%\) with RG decoder for quantum repetition code arranged on a 1D and 2D lattice, respectively [19].

Realizations

NMR: 3-qubit phase-flip code [20,21], with up to two rounds of error correction in liquid-state NMR [22].Trapped ions: 3-qubit bit-flip code by Wineland group [23], and 3-qubit phase-flip algorithm implemented in 3 cycles on high fidelity gate operations [24]. Both phase- and bit-flip codes for 31 qubits and their stabilizer measurements have been realized by Quantinuum [25]. Multiple rounds of Steane error correction [26].Superconducting circuits: 3-qubit phase-flip and bit-flip code by Schoelkopf group [27,28]; 3-qubit bit-flip code [29]; 3-qubit phase-flip code up to 3 cycles of error correction [30]; IBM 15-qubit device [31]; IBM Rochester device using 43-qubit code [32]; Google system performing up to 8 error-correction cycles on 5 and 9 qubits [33]; Google Quantum AI Sycamore utilizing up to 11 physical qubits and running 50 correction rounds [34]; Google Quantum AI Sycamore utilizing up to 25 qubits for comparison of logical error scaling with a quantum code [35] (see also [36]). Google Quantum AI follow-up experiment on codes up to (classical) distance 29, demonstrating exponential suppression to an error floor of \(10^{-10}\) [37]. Ising-model Nishimori phase transition realized for GHZ states on 54 qubits on a 127 qubit IBM device [38]. GHZ state on 75 qubits made on an IBM device [39].Continuous error correction protocols have been implemented on a 3-qubit superconducting qubit device [40].Semiconductor spin-qubit devices: 3-qubit devices at RIKEN [41] and Delft [42].Nitrogen-vacancy centers in diamond: 3-qubit phase-flip code [43,44] (see also Ref. [45]).Repetition codes are used in quantum annealing protocols [46–48].

Notes

Repetition codes can be used to benchmark device performance [49].

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 code — Group-based quantum repetition codes reduce to quantum repetition codes for \(G = \mathbb{Z}_2\).
  • GNU PI code — GNU codewords for \(g=1\) reduce to the phase-flip repetition code.
  • Frustration-free Hamiltonian code — 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 code — The codespace of the quantum repetition code is the ground-state space of a frustration-free classical Ising model with nearest-neighbor interactions.

Cousins

  • Fracton stabilizer code — The 1D quantum repetition code is an ingredient in product constructions that yield several fracton phases [50; Fig. 8].
  • Abelian topological code — The 1D quantum repetition code is an ingredient in product constructions that yield several topological phases [50; Fig. 8].
  • Cluster-state code — GHZ states can be used as resource states for MBQC protocols [51–53].
  • 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 [12,54–57].
  • 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 [58,59].
  • \(D_4\) 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 [60]. This is related to the fact that the four-bit repetition code yields the \(D_4\) hyper-diamond lattice code via Construction A.
  • Concatenated GKP code — Concatenating a three-qubit quantum repetition code with GKP codes can correct some two-bit-flip errors [61] (see also [62]).
  • Coherent-state repetition code — Two-component cat codes in the coherent-state basis have been concatenated with quantum repetition codes [63,64].
  • Self-correcting quantum code — The bit-flip repetition code associated with the 2D classical Ising model is a self-correcting classical memory [65; 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.
  • \([[2^D,D,2]]\) 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 [66].
  • \([[9,1,3]]\) 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 [50] are constructed from a balanced product of the quantum repetion (1D Ising) code and the Newman-Moore code.

References

[1]
A. Peres, “Reversible logic and quantum computers”, Physical Review A 32, 3266 (1985) DOI
[2]
N. D. Mermin, “Quantum mysteries revisited”, American Journal of Physics 58, 731 (1990) DOI
[3]
D. M. Greenberger, M. A. Horne, and A. Zeilinger, “Going Beyond Bell’s Theorem”, (2007) arXiv:0712.0921
[4]
K. Keane and A. N. Korotkov, “Simplified quantum error detection and correction for superconducting qubits”, Physical Review A 86, (2012) arXiv:1205.1836 DOI
[5]
Á. Rozgonyi and G. Széchenyi, “Break-even point of the phase-flip error correcting code”, New Journal of Physics 25, 103004 (2023) arXiv:2303.17810 DOI
[6]
D. Cruz et al., “Efficient Quantum Algorithms for GHZ and W States, and Implementation on the IBM Quantum Computer”, Advanced Quantum Technologies 2, (2019) arXiv:1807.05572 DOI
[7]
S. Bravyi, A. Kliesch, R. Koenig, and E. Tang, “Obstacles to Variational Quantum Optimization from Symmetry Protection”, Physical Review Letters 125, (2020) arXiv:1910.08980 DOI
[8]
A. B. Watts, R. Kothari, L. Schaeffer, and A. Tal, “Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits”, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (2019) arXiv:1906.08890 DOI
[9]
R. Verresen, N. Tantivasadakarn, and A. Vishwanath, “Efficiently preparing Schrödinger’s cat, fractons and non-Abelian topological order in quantum devices”, (2022) arXiv:2112.03061
[10]
C. Ahn, A. C. Doherty, and A. J. Landahl, “Continuous quantum error correction via quantum feedback control”, Physical Review A 65, (2002) arXiv:quant-ph/0110111 DOI
[11]
F. Reiter, A. S. Sørensen, P. Zoller, and C. A. Muschik, “Dissipative quantum error correction and application to quantum sensing with trapped ions”, Nature Communications 8, (2017) arXiv:1702.08673 DOI
[12]
C. Chamberland et al., “Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes”, PRX Quantum 3, (2022) arXiv:2012.04108 DOI
[13]
A. L. Toom, “Nonergodic Multidimensional System of Automata”, Probl. Peredachi Inf., 10:3 (1974), 70–79; Problems Inform. Transmission, 10:3 (1974), 239–246
[14]
L. F. Gray, “Toom’s Stability Theorem in Continuous Time”, Perplexing Problems in Probability 331 (1999) DOI
[15]
P. Gács, Journal of Statistical Physics 103, 45 (2001) arXiv:math/0003117 DOI
[16]
P. Gács, Journal of Statistical Physics 103, 45 (2001) DOI
[17]
B. S. Cirel’son, “Reliable storage of information in a system of unreliable components with local interactions”, Lecture Notes in Mathematics 15 (1978) DOI
[18]
I. Convy, H. Liao, S. Zhang, S. Patel, W. P. Livingston, H. N. Nguyen, I. Siddiqi, and K. B. Whaley, “Machine learning for continuous quantum error correction on superconducting qubits”, New Journal of Physics 24, 063019 (2022) arXiv:2110.10378 DOI
[19]
K. Duivenvoorden, N. P. Breuckmann, and B. M. Terhal, “Renormalization Group Decoder for a Four-Dimensional Toric Code”, IEEE Transactions on Information Theory 65, 2545 (2019) arXiv:1708.09286 DOI
[20]
D. G. Cory, M. D. Price, W. Maas, E. Knill, R. Laflamme, W. H. Zurek, T. F. Havel, and S. S. Somaroo, “Experimental Quantum Error Correction”, Physical Review Letters 81, 2152 (1998) arXiv:quant-ph/9802018 DOI
[21]
O. Moussa, J. Baugh, C. A. Ryan, and R. Laflamme, “Demonstration of Sufficient Control for Two Rounds of Quantum Error Correction in a Solid State Ensemble Quantum Information Processor”, Physical Review Letters 107, (2011) arXiv:1108.4842 DOI
[22]
J. Zhang, D. Gangloff, O. Moussa, and R. Laflamme, “Experimental quantum error correction with high fidelity”, Physical Review A 84, (2011) arXiv:1109.4821 DOI
[23]
J. Chiaverini et al., “Realization of quantum error correction”, Nature 432, 602 (2004) DOI
[24]
P. Schindler, J. T. Barreiro, T. Monz, V. Nebendahl, D. Nigg, M. Chwalla, M. Hennrich, and R. Blatt, “Experimental Repetitive Quantum Error Correction”, Science 332, 1059 (2011) DOI
[25]
S. A. Moses et al., “A Race-Track Trapped-Ion Quantum Processor”, Physical Review X 13, (2023) arXiv:2305.03828 DOI
[26]
L. Postler, F. Butt, I. Pogorelov, C. D. Marciniak, S. Heußen, R. Blatt, P. Schindler, M. Rispler, M. Müller, and T. Monz, “Demonstration of fault-tolerant Steane quantum error correction”, (2023) arXiv:2312.09745
[27]
L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit”, Nature 467, 574 (2010) arXiv:1004.4324 DOI
[28]
M. D. Reed, L. DiCarlo, S. E. Nigg, L. Sun, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Realization of three-qubit quantum error correction with superconducting circuits”, Nature 482, 382 (2012) arXiv:1109.4948 DOI
[29]
D. Ristè, S. Poletto, M.-Z. Huang, A. Bruno, V. Vesterinen, O.-P. Saira, and L. DiCarlo, “Detecting bit-flip errors in a logical qubit using stabilizer measurements”, Nature Communications 6, (2015) arXiv:1411.5542 DOI
[30]
J. Cramer, N. Kalb, M. A. Rol, B. Hensen, M. S. Blok, M. Markham, D. J. Twitchen, R. Hanson, and T. H. Taminiau, “Repeated quantum error correction on a continuously encoded qubit by real-time feedback”, Nature Communications 7, (2016) arXiv:1508.01388 DOI
[31]
J. R. Wootton and D. Loss, “Repetition code of 15 qubits”, Physical Review A 97, (2018) arXiv:1709.00990 DOI
[32]
J. R. Wootton, “Benchmarking near-term devices with quantum error correction”, Quantum Science and Technology 5, 044004 (2020) arXiv:2004.11037 DOI
[33]
J. Kelly et al., “State preservation by repetitive error detection in a superconducting quantum circuit”, Nature 519, 66 (2015) arXiv:1411.7403 DOI
[34]
“Exponential suppression of bit or phase errors with cyclic error correction”, Nature 595, 383 (2021) arXiv:2102.06132 DOI
[35]
R. Acharya et al., “Suppressing quantum errors by scaling a surface code logical qubit”, (2022) arXiv:2207.06431
[36]
K. C. Miao et al., “Overcoming leakage in quantum error correction”, Nature Physics 19, 1780 (2023) arXiv:2211.04728 DOI
[37]
R. Acharya et al., “Quantum error correction below the surface code threshold”, (2024) arXiv:2408.13687
[38]
E. H. Chen et al., “Nishimori transition across the error threshold for constant-depth quantum circuits”, Nature Physics (2024) arXiv:2309.02863 DOI
[39]
H. Liao, G. S. Hartnett, A. Kakkar, A. Tan, M. Hush, P. S. Mundada, M. J. Biercuk, and Y. Baum, “Achieving computational gains with quantum error correction primitives: Generation of long-range entanglement enhanced by error detection”, (2024) arXiv:2411.14638
[40]
W. P. Livingston, M. S. Blok, E. Flurin, J. Dressel, A. N. Jordan, and I. Siddiqi, “Experimental demonstration of continuous quantum error correction”, Nature Communications 13, (2022) arXiv:2107.11398 DOI
[41]
K. Takeda, A. Noiri, T. Nakajima, T. Kobayashi, and S. Tarucha, “Quantum error correction with silicon spin qubits”, Nature 608, 682 (2022) arXiv:2201.08581 DOI
[42]
F. van Riggelen, W. I. L. Lawrie, M. Russ, N. W. Hendrickx, A. Sammak, M. Rispler, B. M. Terhal, G. Scappucci, and M. Veldhorst, “Phase flip code with semiconductor spin qubits”, (2022) arXiv:2202.11530
[43]
G. Waldherr et al., “Quantum error correction in a solid-state hybrid spin register”, Nature 506, 204 (2014) arXiv:1309.6424 DOI
[44]
T. Nakazato, R. Reyes, N. Imaike, K. Matsuda, K. Tsurumoto, Y. Sekiguchi, and H. Kosaka, “Quantum error correction of spin quantum memories in diamond under a zero magnetic field”, Communications Physics 5, (2022) DOI
[45]
T. H. Taminiau, J. Cramer, T. van der Sar, V. V. Dobrovitski, and R. Hanson, “Universal control and error correction in multi-qubit spin registers in diamond”, Nature Nanotechnology 9, 171 (2014) arXiv:1309.5452 DOI
[46]
S. P. Jordan, E. Farhi, and P. W. Shor, “Error-correcting codes for adiabatic quantum computation”, Physical Review A 74, (2006) arXiv:quant-ph/0512170 DOI
[47]
K. L. Pudenz, T. Albash, and D. A. Lidar, “Error-corrected quantum annealing with hundreds of qubits”, Nature Communications 5, (2014) arXiv:1307.8190 DOI
[48]
K. L. Pudenz, T. Albash, and D. A. Lidar, “Quantum annealing correction for random Ising problems”, Physical Review A 91, (2015) arXiv:1408.4382 DOI
[49]
T. Tomesh, P. Gokhale, V. Omole, G. S. Ravi, K. N. Smith, J. Viszlai, X.-C. Wu, N. Hardavellas, M. R. Martonosi, and F. T. Chong, “SupermarQ: A Scalable Quantum Benchmark Suite”, (2022) arXiv:2202.11045
[50]
T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes II. Product constructions”, (2024) arXiv:2402.16831
[51]
M. Gimeno-Segovia, P. Shadbolt, D. E. Browne, and T. Rudolph, “From Three-Photon Greenberger-Horne-Zeilinger States to Ballistic Universal Quantum Computation”, Physical Review Letters 115, (2015) arXiv:1410.3720 DOI
[52]
S.-W. Lee, K. Park, T. C. Ralph, and H. Jeong, “Nearly Deterministic Bell Measurement for Multiphoton Qubits and its Application to Quantum Information Processing”, Physical Review Letters 114, (2015) arXiv:1502.07437 DOI
[53]
S. Omkar, S.-H. Lee, Y. S. Teo, S.-W. Lee, and H. Jeong, “All-Photonic Architecture for Scalable Quantum Computing with Greenberger-Horne-Zeilinger States”, PRX Quantum 3, (2022) arXiv:2109.12280 DOI
[54]
J. Guillaud and M. Mirrahimi, “Repetition Cat Qubits for Fault-Tolerant Quantum Computation”, Physical Review X 9, (2019) arXiv:1904.09474 DOI
[55]
S. Puri et al., “Bias-preserving gates with stabilized cat qubits”, Science Advances 6, (2020) arXiv:1905.00450 DOI
[56]
J. Guillaud and M. Mirrahimi, “Error rates and resource overheads of repetition cat qubits”, Physical Review A 103, (2021) arXiv:2009.10756 DOI
[57]
F.-M. L. Régent, C. Berdou, Z. Leghtas, J. Guillaud, and M. Mirrahimi, “High-performance repetition cat code using fast noisy operations”, Quantum 7, 1198 (2023) arXiv:2212.11927 DOI
[58]
F. Minganti, A. Biella, N. Bartolo, and C. Ciuti, “Spectral theory of Liouvillians for dissipative phase transitions”, Physical Review A 98, (2018) arXiv:1804.11293 DOI
[59]
S. Lieu, R. Belyansky, J. T. Young, R. Lundgren, V. V. Albert, and A. V. Gorshkov, “Symmetry Breaking and Error Correction in Open Quantum Systems”, Physical Review Letters 125, (2020) arXiv:2008.02816 DOI
[60]
B. Royer, S. Singh, and S. M. Girvin, “Encoding Qubits in Multimode Grid States”, PRX Quantum 3, (2022) arXiv:2201.12337 DOI
[61]
K. Fukui, A. Tomita, and A. Okamoto, “Analog Quantum Error Correction with Encoding a Qubit into an Oscillator”, Physical Review Letters 119, (2017) arXiv:1706.03011 DOI
[62]
M. P. Stafford and N. C. Menicucci, “Biased Gottesman-Kitaev-Preskill repetition code”, Physical Review A 108, (2023) arXiv:2212.11397 DOI
[63]
H. Jeong and M. S. Kim, “Efficient quantum computation using coherent states”, Physical Review A 65, (2002) arXiv:quant-ph/0109077 DOI
[64]
T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, “Quantum computation with optical coherent states”, Physical Review A 68, (2003) arXiv:quant-ph/0306004 DOI
[65]
B. J. Brown, D. Loss, J. K. Pachos, C. N. Self, and J. R. Wootton, “Quantum memories at finite temperature”, Reviews of Modern Physics 88, (2016) arXiv:1411.6643 DOI
[66]
D. Hangleiter, M. Kalinowski, D. Bluvstein, M. Cain, N. Maskara, X. Gao, A. Kubica, M. D. Lukin, and M. J. Gullans, “Fault-tolerant compiling of classically hard IQP circuits on hypercubes”, (2024) arXiv:2404.19005
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Zoo Code ID: quantum_repetition

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“Quantum repetition code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_repetition
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@incollection{eczoo_quantum_repetition, title={Quantum repetition code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_repetition} }
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