Qubit code

Qubit code

Alternative names: Qubit subspace code.

Description

Encodes \(K\)-dimensional Hilbert space into a \(2^n\)-dimensional (i.e., \(n\)-qubit) Hilbert space. Usually denoted as \(((n,K))\) or \(((n,K,d))\), where \(d\) is the code's distance.

The qubit codes are equivalent if the codespace of one code can be mapped into that of the other under a tensor product of single-qubit unitary operations and a qubit permutation.

Protection

An \(((n,K,d))\) code corrects erasure errors on up to \(d-1\) qubits. The number of correctable errors is often called the decoding radius, and it is upper bounded by half of the code distance. As a result, qubit codes cannot tolerate adversarial errors on more than \((1-R)/4\) registers, where \(R = \log_2 K/n\) is the code rate.

Pauli-string error basis

A convenient and often considered error set is the Pauli error or Pauli string basis.

Pauli strings: For a single qubit, this set consists of products of powers of the Pauli matrices \begin{align} X=\begin{pmatrix}0 & 1\\ 1 & 0 \end{pmatrix}\,\,\text{ and }\,\,Z=\begin{pmatrix}1 & 0\\ 0 & -1 \end{pmatrix}~. \tag*{(1)}\end{align} For multiple qubits, error set elements are tensor products of elements of the single-qubit error set. Tensor products of \(X\) (\(Z\)) Paulis acting on different qubits are called \(X\)-type (\(Z\)-type) Pauli strings. Combining the \(X\)-type and \(Z\)-type strings with \(i\) forms a group called the Pauli group on \(n\) qubits, while combining them with \(-1\) forms the real Pauli group.

The Pauli error set is a unitary and Hermitian basis for linear operators on the multi-qubit Hilbert space that is orthonormal under the Hilbert-Schmidt inner product; it is a prototypical nice error basis. The distance associated with this set is often the minimum weight of a Pauli string that implements a nontrivial logical operation in the code.

Noise channels

A quantum channel that admits a set of Pauli strings as its Kraus operators is called a Pauli channel, and such channels are typically more tractable than the more general, non-Pauli channels. Relevant Pauli channels include dephasing noise and depolarizing noise (a.k.a. Werner-Holevo channel [1]). Relevant non-Pauli channels are AD noise, erasure (which maps all qubit states into a third state \(|e\rangle\) outside of the qubit Hilbert space), and biased erasure (in which case only the \(|1\rangle\) qubit state is mapped to \(|e\rangle\)). Noise can be correlated in space or in time, with the latter being an example of a non-Markovian phenomenon [2,3].

Quantum weight enumerators and pure distance

Quantum weight enumerator: Determining protection and bounds on code parameters can also be done using the code's Shor-Laflamme quantum weight enumerator [4] (cf. weight enumerators) \begin{align} \begin{split} A(x)&=\sum_{j=0}^{n}A_{j}x^{j}\\ A_{j}&=\frac{1}{K^{2}}\sum_{\text{wt-}j\text{ Paulis }P}\left|\text{tr}(P\Pi)\right|^{2}~, \end{split} \tag*{(2)}\end{align} where \(\Pi\) is the code projection, and where the sum is over the Pauli group modulo the subgroup of phases (hence, the dagger below is necessary in case the coset representative is not Hermitian).

The dual quantum weight enumerator is \begin{align} \begin{split} B(x)&=\sum_{j=0}^{n}B_{j}x^{j}\\ B_{j}&=\frac{1}{K}\sum_{\text{wt-}j\text{ Paulis }P}\text{tr}(P\Pi P^{\dagger}\Pi)~, \end{split} \tag*{(3)}\end{align} and the two satisfy the quantum MacWilliams identity [4]; see [5; Ch. 7]. This identity gives rise to quantum linear programming (LP) bounds [6,7]; see the book [5].

Pure distance: The distance \(d\) of a qubit code is the smallest integer \(0<j=d\) at which the quantum weight enumerator is not equal to its dual, \(A_j \neq B_j\) [8]. A code is called pure if \(A_j = 0\) for all \(0 < j < d\); otherwise, the code is called impure. The pure distance [9,10] (a.k.a. diagonal distance [11,12]) \(d_{\textnormal{pure}}\) is the smallest integer \(1 < j=d_{\textnormal{pure}}\) at which \(A_j > 0\). Codes for which \(d_{\textnormal{pure}} < d\) are impure, otherwise they are pure. For impure codes, there exists a Pauli error of weight less than the \(d\) that has a non-zero expectation value with respect to a code state.

Degenerate qubit codes are impure, but impure codes may not be degenerate [5,13]. There are subtleties with defining degeneracy for non-stabilizer qubit codes with even distance [5].

Other types of quantum weight enumerators are the Rains unitary enumerators [14] and the Rains shadow enumerators [6] (see also [15]), and signed weight enumerators taking into account the sign of the expectation value of a Pauli string [16]. Rains shadow enumerators are related to Bell sampling [17]. These notions can be generalized to qudit codes and other error bases [18–20]. There are techniques to compute them for general codes [20]. Semidefinite programming (SDP) hierarchies and a quantum Delsarte bound have been developed [21].

Rate

Exact two-way assisted capacities have been obtained for the erasure and dephasing channels [22]. There are many bounds on the quantum capacity of the depolarizing channel (e.g., [23]); see review [24].

Transversal Gates

A qubit code is \(U\)-quasi-transversal if it can realize the logical gate \(U\) in the third level of the Clifford hierarchy using the physical gate \(C T^{\otimes n}\), where \(C\) is some Clifford gate [25; Def. 4].

Gates

Clifford group: The Clifford group is the normalizer of the Pauli group. The group consists of the Pauli group as well as elements that permute Pauli operators amongst themselves. Up to any phases and Pauli strings, the group is equivalent to the symplectic group \(Sp(2n,\mathbb{Z}_2)\). See Refs. [5,26–28] for generators, relations, and normal form. The group cannot be expressed as a semidirect product of the Pauli and symplectic groups [29]. There is a canonical form for Clifford circuits [30,31]. Restricting the group to real-valued elements yields the real Clifford group. Single-qubit Clifford gates, together with Paulis, realize a group with \(192\) elements. Modding out phases yields the \(48\)-element \(2O\) binary octahedral subgroup of \(SU(2)\). Further modding out the Pauli group, which corresponds to the quaternion group \(Q\), yields the permutation group \(S_3\), which consists of permutations of the three non-identity single-qubit Pauli matrices. Subgroups of the two-qubit Clifford group have been classified [32].

Computing using Clifford gates only can be efficiently simulated on a classical computer, according to the Gottesman-Knill theorem [33,34]. Universal quantum computing can be achieved using Clifford gates and a single type of non-Clifford gate, such as the \(T\) gate [35]. More generally, the Solovay-Kitaev theorem [36,37] states that any subset of gates the generates a dense subgroup of the full \(n\)-qubit gate group can be used to construct any gate to arbitrary accuracy (see [38][39; Appx. 3]). The task of approximating a desired gate by Clifford gates and a fixed set of non-Clifford gates is called gate compilation or circuit synthesis.Non-Clifford gates are typically more difficult to implement than Clifford gates and so are treated as a resource. Optimizing T-gate count in circuit synthesis is \(NP\)-hard [40] and can be done using various procedures [41–46], e.g., ZX calculus (a.k.a. Penrose spin calculus) [47–50] or reinforcement learning [51]. There is an optimal asymptotic scaling of the number of T gates needed to prepare an arbitrary state [52,53]. Decompositions in terms of Toffoli and Hadamard gates [54] as well as cosine-sine gates also exist [55]. Gate errors in circuit synthesis can sometimes add up destructively [56]. There is a threshold against depolarizing noise for any single-qubit gate that determines if the gate enables universal quantum computation [57,58].

Clifford hierarchy: The Clifford hierarchy [59–63] is a tower of gate sets which includes Pauli and Clifford gates at its first two levels, and non-Clifford gates at higher levels. The \(k\)th level is defined recursively by \begin{align} C_k = \{ U | U P U^{\dagger} \in C_{k-1} \}~, \tag*{(4)}\end{align} where \(P\) is any Pauli matrix, where \(C_1\) is the Pauli group, and where \(C_2\) is the Clifford group. Gates for one qubit have been classified [64].

Arbitrary \(n\)-qubit circuits can be implemented fault-tolerantly in a 3D architecture using \(O(n^{3/2}\log^3 n)\) qubits, and in a 2D architecture using only \(O(n^2 \log^3 n)\) qubits [65].Fault-tolerant gates can be done for any code supporting a transversal implementation of Pauli gates using generalized gate teleportation [66].

Decoding

Syndrome measurements are assumed to be perfect in the code-capacity model. Incorporating faulty syndrome measurements can be done using the phenomenological noise model, which simulates errors during syndrome extraction by flipping some of the bits of the measured syndrome bitstring. In the more involved circuit-level noise model, every component of the syndrome extraction circuit can be faulty.The decoder determining the most likely error given a noise channel is called the maximum probability error (MPE) decoder. For few-qubit codes (\(n\) is small), MPE decoding can be based by creating a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used.

Effective distance and hook errors: Decoders are characterized by an effective distance (a.k.a. circuit-level distance), the minimum number of faulty operations during syndrome measurement that is required to make an undetectable error. A code is distance-preserving if it admits a decoder whose circuit-level distance is equal to the code distance. A particularly dangerous class of syndrome measurement circuit faults are hook errors, which are ancilla faults that cause more than one data-qubit error [67]. Hook errors occur at specific places in a syndrome extraction circuit and can sometimes be removed by re-ordering the gates of the circuit. If not, the use of flag qubits (see [5]) to detect hook errors may be necessary to yield fault-tolerant decoders.

Fault Tolerance

There are lower bounds on the overhead of fault-tolerant QEC in terms of the capacity of the noise channel [68]. A more stringent bound applies to geometrically local QEC due to the fact that locality constrains the growth of the entanglement that is needed for protection [69].Arbitrary \(n\)-qubit circuits can be implemented fault-tolerantly in a 3D architecture using \(O(n^{3/2}\log^3 n)\) qubits, and in a 2D architecture using only \(O(n^2 \log^3 n)\) qubits [65].Fault-tolerant gates can be done for any code supporting a transversal implementation of Pauli gates using generalized gate teleportation [66].

Threshold

Computational threshold: A fault-tolerant computational threshold is the maximum noise rate in a particular single-parameter noise model below which any logical computation of size \(M\) can be executed on a physical-qubit architecture to arbitrary accuracy and with an overhead of order \(O(M\text{polylog}M)\). The first methods to achieve a computational threshold use recursively concatenated stabilizer code families [70–77]; such a threshold is called a concatenated threshold. Initially proven under local stochastic noise, the concatenated threshold theorem also holds for various types of non-Markovian noise [75,76,78,79] and leakage errors [80]. The resulting concatenated code is highly degenerate, with all but an exponentially small fraction of generators having small weights. Circuit and measurement designs have to take care of the few stabilizer generators with large weights in order to be fault tolerant, but measurement duration may not pose a threat to scalability [81]. Concatenated methods require constant-space and polylogarithmic-time overhead, but concatenations using quantum Hamming codes improve this to quasi-polylogarithmic time [82,83], and concatenations of the Steane code and certain QLDPC codes further improve this to polylogarithmic time [84]. Subsequently, thresholds were determined for infinite families of lattice stabilizer codes, starting with the toric code [67]; such a threshold is colloquially called a topological threshold. Fault-tolerant computations with no notion of locality can be made local on a 2D or 3D geometry with minimal overhead [65].

There is a threshold against depolarizing noise for any single-qubit gate that determines if the gate enables universal quantum computation [57,58].

Measurement threshold: One can derive conditions quantifying how many random single-qubit measurements can be made without destroying the logical information [85]. The measurement threshold is the maximum total probability that a single qubit is measured in a random \(X\), \(Y\), or \(Z\) basis at which the logical information is still recoverable. The measurement threshold is at least as large as the erasure threshold [85; Thm. 4].

There is a dynamical phase transition between a bounded-error and an unbounded-error phase for a model of qubits weakly coupled to a refrigerator [86].

Notes

There is a relation between one-way entanglement distillation protocols and QECCs [87].Qubit error correction is required for unconditionally secure quantum key distribution [88].See Qiskit QEC framework for realizing protocols on IBM machines.There exists a distance- and rate-dependent lower bound on the degree of entanglement of a qubit code [89; Thm. 3i].

Cousins

Gray code— Gray codes are useful for optimizing qubit unitary circuits [90] and for encoding qudits in multiple qubits [91].
Reed-Muller (RM) code— Optimizing T gates in a qubit circuit that uses CNOT and T gates is equivalent to decoding a particular RM code [43].
EA qubit code— EA qubit codes utilize additional ancillary qubits in a pre-shared entangled state, but reduce to ordinary qubit codes when said qubits are interpreted as noiseless physical qubits.
Hybrid qubit code— A hybrid qubit code storing no classical information reduces to a qubit code. Conversely, any qubit code can be converted into a hybrid qubit code by using some its qubits to store only classical information [92].
Five-qubit perfect code— Every \(((5,2,3))\) code is single-qubit-Clifford-equivalent to the five-qubit code [93; Corr. 10].
Subsystem qubit code— Subsystem qubit codes reduce to (subspace) qubit codes when there is no gauge subsystem.

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Parents

Operator-algebra (OA) qubit codeQuantum

An OA qubit code which has no gauge qubits and no block structure is a qubit code.

Modular-qudit codeQECC Quantum

Modular-qudit quantum codes for \(q=2\) correspond to qubit codes.

Galois-qudit codeQECC Quantum

Galois-qudit quantum codes for \(q=2\) correspond to qubit codes.

Spin codeQECC Quantum

Spin codes with spin \(\ell=1/2\) correspond to qubit codes.

Qubit code

Children

Dynamical automorphism (DA) codeFloquet

Haar-random qubit code

Local Haar-random circuit qubit code

Conformal-field theory (CFT) code

Kim-Preskill-Tang (KPT) code

Fermion code

The Majorana operator algebra is isomorphic to the qubit Pauli-operator algebra via various fermion-into-qubit encodings.

Circuit-to-Hamiltonian approximate code

Eigenstate thermalization hypothesis (ETH) code

ETH codewords are eigenstates of a local Hamiltonian whose eigenstates satisfy ETH.

Jump code

Movassagh-Ouyang Hamiltonian codeQuantum Reed-Muller Color BB Homological Kitaev surface

Self-complementary qubit code

Union stabilizer (USt) codeBB Homological Fermion-into-qubit Quantum Reed-Muller Color Twist-defect color CDSC Twist-defect surface Kitaev surface

XP stabilizer codeBB Homological Fermion-into-qubit Quantum Reed-Muller Color Twist-defect color CDSC Twist-defect surface Kitaev surface

PI qubit code

Concatenated qubit code

Neural network quantum code

Numerically optimized four-qubit AD code

\(((8,1,3))\) Plenio-Vedral-Knight CE code

Cubic theory code

Chen-Hsin invertible-order code

References

[1]: R. F. Werner and A. S. Holevo, “Counterexample to an additivity conjecture for output purity of quantum channels”, Journal of Mathematical Physics 43, 4353 (2002) arXiv:quant-ph/0203003 DOI
[2]: R. Klesse and S. Frank, “Quantum Error Correction in Spatially Correlated Quantum Noise”, Physical Review Letters 95, (2005) arXiv:quant-ph/0505153 DOI
[3]: S. Milz and K. Modi, “Quantum Stochastic Processes and Quantum non-Markovian Phenomena”, PRX Quantum 2, (2021) arXiv:2012.01894 DOI
[4]: P. Shor and R. Laflamme, “Quantum MacWilliams Identities”, (1996) arXiv:quant-ph/9610040
[5]: D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
[6]: E. M. Rains, “Quantum shadow enumerators”, (1997) arXiv:quant-ph/9611001
[7]: A. Ashikhmin and S. Litsyn, “Upper Bounds on the Size of Quantum Codes”, (1997) arXiv:quant-ph/9709049
[8]: A. Ashikhmin, A. Barg, E. Knill, and S. Litsyn, “Quantum Error Detection I: Statement of the Problem”, (1999) arXiv:quant-ph/9906126
[9]: Y. Fujiwara, “Ability of stabilizer quantum error correction to protect itself from its own imperfection”, Physical Review A 90, (2014) arXiv:1409.2559 DOI
[10]: T. Wagner, H. Kampermann, D. Bruß, and M. Kliesch, “Pauli channels can be estimated from syndrome measurements in quantum error correction”, Quantum 6, 809 (2022) arXiv:2107.14252 DOI
[11]: S. Y. Looi, L. Yu, V. Gheorghiu, and R. B. Griffiths, “Quantum-error-correcting codes using qudit graph states”, Physical Review A 78, (2008) arXiv:0712.1979 DOI
[12]: U. S. Kapshikar, “The Diagonal Distance of CWS Codes”, (2021) arXiv:2107.11286
[13]: A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
[14]: E. M. Rains, “Quantum Weight Enumerators”, (1996) arXiv:quant-ph/9612015
[15]: A. J. Scott, “Probabilities of Failure for Quantum Error Correction”, Quantum Information Processing 4, 399 (2005) arXiv:quant-ph/0406063 DOI
[16]: P. Rall, “Signed quantum weight enumerators characterize qubit magic state distillation”, (2017) arXiv:1702.06990
[17]: D. Miller et al., “Experimental measurement and a physical interpretation of quantum shadow enumerators”, (2024) arXiv:2408.16914
[18]: C. Hu, S. Yang, and S. S.-T. Yau, “Weight enumerators for nonbinary asymmetric quantum codes and their applications”, Advances in Applied Mathematics 121, 102085 (2020) DOI
[19]: C. Cao and B. Lackey, “Quantum weight enumerators and tensor networks”, (2023) arXiv:2211.02756
[20]: C. Cao, M. J. Gullans, B. Lackey, and Z. Wang, “Quantum Lego Expansion Pack: Enumerators from Tensor Networks”, (2024) arXiv:2308.05152
[21]: G. A. Munné, A. Nemec, and F. Huber, “SDP bounds on quantum codes”, (2024) arXiv:2408.10323
[22]: S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental limits of repeaterless quantum communications”, Nature Communications 8, (2017) arXiv:1510.08863 DOI
[23]: G. Smith, J. A. Smolin, and A. Winter, “The Quantum Capacity With Symmetric Side Channels”, IEEE Transactions on Information Theory 54, 4208 (2008) arXiv:quant-ph/0607039 DOI
[24]: L. Gyongyosi, S. Imre, and H. V. Nguyen, “A Survey on Quantum Channel Capacities”, IEEE Communications Surveys & Tutorials 20, 1149 (2018) arXiv:1801.02019 DOI
[25]: E. T. Campbell and M. Howard, “Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost”, Physical Review A 95, (2017) arXiv:1606.01904 DOI
[26]: J. Dehaene and B. De Moor, “Clifford group, stabilizer states, and linear and quadratic operations over GF(2)”, Physical Review A 68, (2003) arXiv:quant-ph/0304125 DOI
[27]: M. V. den Nest, “Classical simulation of quantum computation, the Gottesman-Knill theorem, and slightly beyond”, (2009) arXiv:0811.0898
[28]: P. Selinger, “Generators and relations for n-qubit Clifford operators”, Logical Methods in Computer Science Volume 11, Issue 2, (2015) arXiv:1310.6813 DOI
[29]: S. Burton, E. Durso-Sabina, and N. C. Brown, “Genons, Double Covers and Fault-tolerant Clifford Gates”, (2024) arXiv:2406.09951
[30]: S. Bravyi and D. Maslov, “Hadamard-Free Circuits Expose the Structure of the Clifford Group”, IEEE Transactions on Information Theory 67, 4546 (2021) arXiv:2003.09412 DOI
[31]: D. Ostrev, “Canonical Form and Finite Blocklength Bounds for Stabilizer Codes”, (2024) arXiv:2408.15202
[32]: E. Kubischta and I. Teixeira, “Classification of the Subgroups of the Two-Qubit Clifford Group”, (2024) arXiv:2409.14624
[33]: D. Gottesman, “The Heisenberg Representation of Quantum Computers”, (1998) arXiv:quant-ph/9807006
[34]: E. Knill, private communication, ca. 1998.
[35]: A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, “Elementary gates for quantum computation”, Physical Review A 52, 3457 (1995) arXiv:quant-ph/9503016 DOI
[36]: A. Y. Kitaev, “Quantum computations: algorithms and error correction”, Russian Mathematical Surveys 52, 1191 (1997) DOI
[37]: A. Kitaev, A. Shen, and M. Vyalyi, Classical and Quantum Computation (American Mathematical Society, 2002) DOI
[38]: C. M. Dawson and M. A. Nielsen, “The Solovay-Kitaev algorithm”, (2005) arXiv:quant-ph/0505030
[39]: “The Solovay–Kitaev theorem”, Quantum Computation and Quantum Information 617 (2012) DOI
[40]: J. van de Wetering and M. Amy, “Optimising quantum circuits is generally hard”, (2024) arXiv:2310.05958
[41]: M. Amy, D. Maslov, and M. Mosca, “Polynomial-Time T-Depth Optimization of Clifford+T Circuits Via Matroid Partitioning”, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 33, 1476 (2014) arXiv:1303.2042 DOI
[42]: D. Gosset, V. Kliuchnikov, M. Mosca, and V. Russo, “An algorithm for the T-count”, (2013) arXiv:1308.4134
[43]: M. Amy and M. Mosca, “T-Count Optimization and Reed–Muller Codes”, IEEE Transactions on Information Theory 65, 4771 (2019) arXiv:1601.07363 DOI
[44]: Y. Nam, N. J. Ross, Y. Su, A. M. Childs, and D. Maslov, “Automated optimization of large quantum circuits with continuous parameters”, npj Quantum Information 4, (2018) arXiv:1710.07345 DOI
[45]: L. Heyfron and E. T. Campbell, “An Efficient Quantum Compiler that reduces \(T\) count”, (2018) arXiv:1712.01557
[46]: V. Gheorghiu, M. Mosca, and P. Mukhopadhyay, “T-count and T-depth of any multi-qubit unitary”, npj Quantum Information 8, (2022) arXiv:2110.10292 DOI
[47]: A. Kissinger and J. van de Wetering, “Reducing the number of non-Clifford gates in quantum circuits”, Physical Review A 102, (2020) arXiv:1903.10477 DOI
[48]: N. de Beaudrap, X. Bian, and Q. Wang, “Techniques to Reduce π/4-Parity-Phase Circuits, Motivated by the ZX Calculus”, Electronic Proceedings in Theoretical Computer Science 318, 131 (2020) arXiv:1911.09039 DOI
[49]: N. de Beaudrap, X. Bian, and Q. Wang, “Fast and Effective Techniques for T-Count Reduction via Spider Nest Identities”, (2020) arXiv:2004.05164 DOI
[50]: A. Kissinger and J. van de Wetering, “Simulating quantum circuits with ZX-calculus reduced stabiliser decompositions”, Quantum Science and Technology 7, 044001 (2022) arXiv:2109.01076 DOI
[51]: F. J. R. Ruiz et al., “Quantum Circuit Optimization with AlphaTensor”, (2024) arXiv:2402.14396
[52]: G. H. Low, V. Kliuchnikov, and L. Schaeffer, “Trading T gates for dirty qubits in state preparation and unitary synthesis”, Quantum 8, 1375 (2024) arXiv:1812.00954 DOI
[53]: D. Gosset, R. Kothari, and K. Wu, “Quantum state preparation with optimal T-count”, (2024) arXiv:2411.04790
[54]: Y. Shi, “Both Toffoli and Controlled-NOT need little help to do universal quantum computation”, (2002) arXiv:quant-ph/0205115
[55]: M. Möttönen, J. J. Vartiainen, V. Bergholm, and M. M. Salomaa, “Quantum Circuits for General Multiqubit Gates”, Physical Review Letters 93, (2004) arXiv:quant-ph/0404089 DOI
[56]: M. B. Hastings, “Turning Gate Synthesis Errors into Incoherent Errors”, (2016) arXiv:1612.01011
[57]: W. van Dam and M. Howard, “Tight Noise Thresholds for Quantum Computation with Perfect Stabilizer Operations”, Physical Review Letters 103, (2009) arXiv:0907.3189 DOI
[58]: W. van Dam and M. Howard, “Noise thresholds for higher-dimensional systems using the discrete Wigner function”, Physical Review A 83, (2011) arXiv:1011.2497 DOI
[59]: D. Gottesman and I. L. Chuang, “Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations”, Nature 402, 390 (1999) arXiv:quant-ph/9908010 DOI
[60]: S. X. Cui, D. Gottesman, and A. Krishna, “Diagonal gates in the Clifford hierarchy”, Physical Review A 95, (2017) arXiv:1608.06596 DOI
[61]: N. Rengaswamy, R. Calderbank, and H. D. Pfister, “Unifying the Clifford hierarchy via symmetric matrices over rings”, Physical Review A 100, (2019) arXiv:1902.04022 DOI
[62]: J. T. Anderson, “On Groups in the Qubit Clifford Hierarchy”, Quantum 8, 1370 (2024) arXiv:2212.05398 DOI
[63]: Z. He, L. Robitaille, and X. Tan, “Permutation gates in the third level of the Clifford hierarchy”, (2024) arXiv:2410.11818
[64]: N. de Silva and O. Lautsch, “The Clifford hierarchy for one qubit or qudit”, (2025) arXiv:2501.07939
[65]: S. H. Choe and R. Koenig, “How to fault-tolerantly realize any quantum circuit with local operations”, (2024) arXiv:2402.13863
[66]: E. Kubischta and I. Teixeira, “Flexible Fault Tolerant Gate Gadgets”, (2024) arXiv:2409.11616
[67]: E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
[68]: O. Fawzi, A. Müller-Hermes, and A. Shayeghi, “A Lower Bound on the Space Overhead of Fault-Tolerant Quantum Computation”, (2022) arXiv:2202.00119 DOI
[69]: N. Baspin, O. Fawzi, and A. Shayeghi, “A lower bound on the overhead of quantum error correction in low dimensions”, (2023) arXiv:2302.04317
[70]: E. Knill, R. Laflamme, and W. H. Zurek, “Resilient quantum computation: error models and thresholds”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 365 (1998) arXiv:quant-ph/9702058 DOI
[71]: J. Preskill, “Reliable quantum computers”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 385 (1998) arXiv:quant-ph/9705031 DOI
[72]: D. Gottesman, “Fault-tolerant quantum computation with local gates”, Journal of Modern Optics 47, 333 (2000) arXiv:quant-ph/9903099 DOI
[73]: D. Aharonov and M. Ben-Or, “Fault-Tolerant Quantum Computation With Constant Error Rate”, (1999) arXiv:quant-ph/9906129
[74]: K. M. Svore, B. M. Terhal, and D. P. DiVincenzo, “Local fault-tolerant quantum computation”, Physical Review A 72, (2005) arXiv:quant-ph/0410047 DOI
[75]: P. Aliferis, D. Gottesman, and J. Preskill, “Quantum accuracy threshold for concatenated distance-3 codes”, (2005) arXiv:quant-ph/0504218
[76]: P. Aliferis, “Level Reduction and the Quantum Threshold Theorem”, (2011) arXiv:quant-ph/0703230
[77]: K. M. Svore, D. P. DiVincenzo, and B. M. Terhal, “Noise Threshold for a Fault-Tolerant Two-Dimensional Lattice Architecture”, (2006) arXiv:quant-ph/0604090
[78]: B. M. Terhal and G. Burkard, “Fault-tolerant quantum computation for local non-Markovian noise”, Physical Review A 71, (2005) arXiv:quant-ph/0402104 DOI
[79]: D. Aharonov, A. Kitaev, and J. Preskill, “Fault-Tolerant Quantum Computation with Long-Range Correlated Noise”, Physical Review Letters 96, (2006) arXiv:quant-ph/0510231 DOI
[80]: P. Aliferis and B. M. Terhal, “Fault-Tolerant Quantum Computation for Local Leakage Faults”, (2006) arXiv:quant-ph/0511065
[81]: D. P. DiVincenzo and P. Aliferis, “Effective Fault-Tolerant Quantum Computation with Slow Measurements”, Physical Review Letters 98, (2007) arXiv:quant-ph/0607047 DOI
[82]: H. Yamasaki and M. Koashi, “Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation”, Nature Physics 20, 247 (2024) arXiv:2207.08826 DOI
[83]: S. Yoshida, S. Tamiya, and H. Yamasaki, “Concatenate codes, save qubits”, (2024) arXiv:2402.09606
[84]: S. Tamiya, M. Koashi, and H. Yamasaki, “Polylog-time- and constant-space-overhead fault-tolerant quantum computation with quantum low-density parity-check codes”, (2024) arXiv:2411.03683
[85]: D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
[86]: M. Bilokur, S. Gopalakrishnan, and S. Majidy, “Thermodynamic limitations on fault-tolerant quantum computing”, (2024) arXiv:2411.12805
[87]: C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction”, Physical Review A 54, 3824 (1996) arXiv:quant-ph/9604024 DOI
[88]: H.-K. Lo and H. F. Chau, “Unconditional Security of Quantum Key Distribution over Arbitrarily Long Distances”, Science 283, 2050 (1999) arXiv:quant-ph/9803006 DOI
[89]: S. Bravyi, D. Lee, Z. Li, and B. Yoshida, “How much entanglement is needed for quantum error correction?”, (2024) arXiv:2405.01332
[90]: J. J. Vartiainen, M. Möttönen, and M. M. Salomaa, “Efficient Decomposition of Quantum Gates”, Physical Review Letters 92, (2004) arXiv:quant-ph/0312218 DOI
[91]: N. P. D. Sawaya, T. Menke, T. H. Kyaw, S. Johri, A. Aspuru-Guzik, and G. G. Guerreschi, “Resource-efficient digital quantum simulation of d-level systems for photonic, vibrational, and spin-s Hamiltonians”, npj Quantum Information 6, (2020) arXiv:1909.12847 DOI
[92]: I. Kremsky, M.-H. Hsieh, and T. A. Brun, “Classical enhancement of quantum-error-correcting codes”, Physical Review A 78, (2008) arXiv:0802.2414 DOI
[93]: E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043

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Victor V. Albert (2023-01-08) — most recent
Sam Gunn (2022-01-08)
Victor V. Albert (2022-05-07)
Victor V. Albert (2021-10-29)

Cite as:

“Qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/qubits_into_qubits

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/qubits_into_qubits.yml.

Qubit code

Description

Protection

Pauli-string error basis

Noise channels

Quantum weight enumerators and pure distance

Rate

Transversal Gates

Gates

Decoding

Fault Tolerance

Threshold

Notes

Cousins

Member of code lists

Primary Hierarchy

References

Your contribution is welcome!

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