Qubit code 

Root code for the Qubit Kingdom

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.

Protection

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.

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

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 [13]. The distance associated with this set is often the minimum weight of a Pauli string that implements a nontrivial logical operation in the code.

The minimum weight of a Pauli error that has a non-zero expectation value for some code basis state is called the diagonal distance [4]. Codes whose distance is greater than the diagonal distance are degenerate.

A quantum channel whose Kraus operators are Pauli strings is called a Pauli channel, and such channels are typically more tractable than general, non-Pauli channels. Relevant Pauli channels include dephasing noise and depolarizing noise (a.k.a. Werner-Holevo channel [5]), while relevant non-Pauli channels are amplitude damping, 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 \(|e\rangle\)).

Bounds on code performance include the quantum Singleton bound [68], quantum Hamming bound [9], and quantum Gilbert-Varshamov bound [9]. Linear programming bounds also exist [10,11].

Determining protection and bounds on code parameters can also be done using Shor-Laflamme quantum weight enumerators [12] and Rains shadow enumerators [10] (see also [13]).

Rate

Exact two-way assisted capacities have been obtained for the erasure and dephasing channels [14].

Gates

Universal computing can be achieved using Clifford gates and a single type of non-Clifford gate, such as the \(T\) gate [15]. Non-Clifford gates are typically more difficult to implement than Clifford gates and so are treated as a resource. Optimizing T-gate count is \(NP\)-hard [16].

Decoding

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 bit string. In the more involved circuit-level noise model, every component of the syndrome extraction circuit can be faulty.Hook errors are syndrome measurement circuit faults that cause more than one data-qubit error [17]. 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 to detect hook errors may be necessary to yield fault-tolerant decoders.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.Decoders are characterized by an effective distance or 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.

Fault Tolerance

There are lower bounds on the overhead of fault-tolerant QEC in terms of the capacity of the noise channel [18]. 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 [19].

Threshold

Measurement threshold: One can derive conditions quantifying how many random single-qubit measurements can be made without destroying the logical information [20]. 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 [20; Thm. 4].

Notes

There is a relation between one-way entanglement distillation protocols and QECCs [21].See Qiskit QEC framework for realizing protocols on IBM machines.

Parents

  • Modular-qudit code — Modular-qudit quantum codes for \(q=2\) correspond to qubit codes.
  • Galois-qudit code — Galois-qudit quantum codes for \(q=2\) correspond to qubit codes.
  • Spin code — Spin codes with spin \(\ell=1/2\) correspond to qubit codes.

Children

Cousins

  • Qubit c-q code — Qubit c-q codes are qubit codes designed to transmit classical information.
  • Fock-state bosonic code — Fock-state code whose codewords are finite superpositions of Fock states with maximum occupation \(N\) can be mapped into a qubit code with \(n\geq\log_2 N\) by performing a binary expansion of the Fock-state labels \(n\) and treating each binary digit as an index for a qubit state. Pauli operators for the constituent qubits can be expressed in terms of bosonic raising and lowering operators [23]. However, noise models for the two code families induce different notions of locality and thus qualitatively different physical interpretations [24].
  • 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.
  • Five-qubit perfect code — Every \(((5,2,3))\) code is equivalent to the five-qubit code [25; Corr. 10].
  • Subsystem qubit code — Subsystem qubit codes reduce to (subspace) qubit codes when there is no gauge subsystem.
  • Single-spin code — Certain single-spin codes yield permutation-invariant qubit codes with non-trivial distance [26] when the single spin is treated as a collective spin of several qubits.

References

[1]
E. Knill, “Non-binary Unitary Error Bases and Quantum Codes”, (1996) arXiv:quant-ph/9608048
[2]
E. Knill, “Group Representations, Error Bases and Quantum Codes”, (1996) arXiv:quant-ph/9608049
[3]
A. Klappenecker and M. Roetteler, “Beyond Stabilizer Codes I: Nice Error Bases”, (2001) arXiv:quant-ph/0010082
[4]
U. S. Kapshikar, “The Diagonal Distance of CWS Codes”, (2021) arXiv:2107.11286
[5]
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
[6]
E. Knill, R. Laflamme, and L. Viola, “Theory of Quantum Error Correction for General Noise”, Physical Review Letters 84, 2525 (2000) arXiv:quant-ph/9604034 DOI
[7]
N. J. Cerf and R. Cleve, “Information-theoretic interpretation of quantum error-correcting codes”, Physical Review A 56, 1721 (1997) arXiv:quant-ph/9702031 DOI
[8]
E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
[9]
A. Ekert and C. Macchiavello, “Error Correction in Quantum Communication”, (1996) arXiv:quant-ph/9602022
[10]
E. M. Rains, “Quantum shadow enumerators”, (1997) arXiv:quant-ph/9611001
[11]
A. Ashikhmin and S. Litsyn, “Upper Bounds on the Size of Quantum Codes”, (1997) arXiv:quant-ph/9709049
[12]
P. Shor and R. Laflamme, “Quantum MacWilliams Identities”, (1996) arXiv:quant-ph/9610040
[13]
A. J. Scott, “Probabilities of Failure for Quantum Error Correction”, Quantum Information Processing 4, 399 (2005) arXiv:quant-ph/0406063 DOI
[14]
S. Pirandola et al., “Fundamental limits of repeaterless quantum communications”, Nature Communications 8, (2017) arXiv:1510.08863 DOI
[15]
A. Barenco et al., “Elementary gates for quantum computation”, Physical Review A 52, 3457 (1995) arXiv:quant-ph/9503016 DOI
[16]
J. van de Wetering and M. Amy, “Optimising T-count is NP-hard”, (2023) arXiv:2310.05958
[17]
E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
[18]
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
[19]
N. Baspin, O. Fawzi, and A. Shayeghi, “A lower bound on the overhead of quantum error correction in low dimensions”, (2023) arXiv:2302.04317
[20]
D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
[21]
C. H. Bennett et al., “Mixed-state entanglement and quantum error correction”, Physical Review A 54, 3824 (1996) arXiv:quant-ph/9604024 DOI
[22]
A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires”, Physics-Uspekhi 44, 131 (2001) arXiv:cond-mat/0010440 DOI
[23]
Victor V. Albert, private communication, 2016
[24]
S. M. Girvin, “Introduction to quantum error correction and fault tolerance”, SciPost Physics Lecture Notes (2023) arXiv:2111.08894 DOI
[25]
E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
[26]
S. Omanakuttan and J. A. Gross, “Multispin Clifford codes for angular momentum errors in spin systems”, Physical Review A 108, (2023) arXiv:2304.08611 DOI
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Zoo Code ID: qubits_into_qubits

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“Qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/qubits_into_qubits
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@incollection{eczoo_qubits_into_qubits, title={Qubit code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qubits_into_qubits} }
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“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.