Qubit code 

Also known as Qubit subspace 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.

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.

The minimum weight of a Pauli error that has a non-zero expectation value for some code basis state is called the diagonal distance [1]. Codes whose distance is greater than the diagonal distance are degenerate. Degenerate codes admit undetectable Pauli errors (i.e., errors whose projection into the codespace is nonzero) of weight less than the code distance (i.e., the projection satisfies the Knill-Laflamme conditions).

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 [2]). 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 [3,4].

Quantum weight enumerators

Quantum weight enumerator: Determining protection and bounds on code parameters can also be done using the code's Shor-Laflamme quantum weight enumerator [5] (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} The weight enumerator and its dual satisfy the quantum MacWilliams identity [5]; see [6; Ch. 7]. It gives rise to quantum linear programming (LP) bounds [7,8]; see the book [6].

The distance \(d\) of a code is the smallest \(j=d\) at which \(A_j \neq B_j\) [9]. Such a code is called pure if \(A_j = B_j = 0\) for all \(j < d\); otherwise, the code is called impure. Degeneracy is sufficient but not necessary for impurity [6].

Other types of quantum weight enumerators are the Rains unitary enumerators [10] and the Rains shadow enumerators [7] (see also [11]), with the latter related to Bell sampling [12]. These notions can be generalized to qudit codes and other error bases [13–15]. There are techniques to compute them for general codes [15]. Semidefinite programming (SDP) hierarchies and a quantum Delsarte bound have been developed [16].

Rate

Exact two-way assisted capacities have been obtained for the erasure and dephasing channels [17].

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 [18; 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. [6,19–21] for generators, relations, and normal form. The group cannot be expressed as a semidirect product of the Pauli and symplectic groups [22]. There is a canonical form for Clifford circuits [23,24]. 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 [25].

Computing using Clifford gates only can be efficiently simulated on a classical computer, according to the Gottesman-Knill theorem [26,27]. Universal quantum computing can be achieved using Clifford gates and a single type of non-Clifford gate, such as the \(T\) gate [28]. More generally, the Solovay-Kitaev theorem [29,30] 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 [31][32; 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 [33] and can be done using various procedures [34–39], e.g., ZX calculus (a.k.a. Penrose spin calculus) [40–43] or reinforcement learning [44]. There is an optimal asymptotic scaling of the number of T gates needed to prepare an arbitrary state [45,46]. Decompositions in terms of Toffoli and Hadamard gates [47] as well as cosine-sine gates also exist [48]. Gate errors in circuit synthesis can sometimes add up destructively [49].

Clifford hierarchy: The Clifford hierarchy [50–54] 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.

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 [55].Fault-tolerant gates can be done for any code supporting a transversal implementation of Pauli gates using generalized gate teleportation [56].

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 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 faults that cause more than one data-qubit error [57]. 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 [6]) 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 [58]. 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 [59].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 [55].Fault-tolerant gates can be done for any code supporting a transversal implementation of Pauli gates using generalized gate teleportation [56].

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 [60–66]; such a threshold is called a concatenated threshold. Such methods require constant-space and polylogarithmic-time overhead, but concatenations using quantum Hamming codes improve this to quasi-polylogarithmic time [67]. Subsequently, thresholds were determined for infinite families of lattice stabilizer codes, starting with the toric code [57]; 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 [55].

Measurement threshold: One can derive conditions quantifying how many random single-qubit measurements can be made without destroying the logical information [68]. 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 [68; Thm. 4].

Notes

There is a relation between one-way entanglement distillation protocols and QECCs [69].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 [70; Thm. 3i].

Parents

Children

Cousins

  • Gray code — Gray codes are useful for optimizing qubit unitary circuits [71] and for encoding qudits in multiple qubits [72].
  • 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 [36].
  • 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 [73].
  • Five-qubit perfect code — Every \(((5,2,3))\) code is single-qubit-Clifford-equivalent to the five-qubit code [74; Corr. 10].
  • Subsystem qubit code — Subsystem qubit codes reduce to (subspace) qubit codes when there is no gauge subsystem.

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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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