\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code[1]

Cousins

• \([2^r-1,2^r-r-1,3]\) Hamming code— Quantum Hamming codes result from applying the CSS construction to Hamming codes and their duals the simplex codes.
• \([2^m-1,m,2^{m-1}]\) simplex code— Quantum Hamming codes result from applying the CSS construction to Hamming codes and their duals the simplex codes.
• Concatenated qubit code— Concatenating a growing sequence of quantum Hamming codes yields fault-tolerant quantum computation with constant space overhead and quasi-polylogarithmic time overhead [2]. Concatenating quantum Hamming codes on top of the \([[4,2,2]]\) and \(C_6\) codes yields fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [4]. In the optimized protocol of Ref. [4], a level-five \(C_4/C_6\) code underlies concatenated quantum Hamming codes \(\mathcal{Q}_5,\mathcal{Q}_6,\mathcal{Q}_7,\mathcal{Q}_7\), yielding a \(2.5\%\) threshold and space overheads \(162\) and \(373\) physical qubits per logical qubit at physical error rate \(0.1\%\) for logical CNOT error rates \(10^{-10}\) and \(10^{-24}\), respectively. A modified tower of interleaved quantum Hamming codes with reserved qubits and recursive hookless Pauli-product measurements yields fault-tolerant quantum computation on a 1D nearest-neighbor qubit line with asymptotic rate above \(5\%\), constant space overhead, quasi-polylogarithmic time overhead, and a threshold [5]. Quantum Hamming codes can also be concatenated with surface codes [6].
• \([[4,2,2]]\) Four-qubit code— Concatenating quantum Hamming codes on top of the \([[4,2,2]]\) and \(C_6\) codes yields fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [4]. In the optimized protocol of Ref. [4], a level-five \(C_4/C_6\) code underlies concatenated quantum Hamming codes \(\mathcal{Q}_5,\mathcal{Q}_6,\mathcal{Q}_7,\mathcal{Q}_7\), yielding a \(2.5\%\) threshold and space overheads \(162\) and \(373\) physical qubits per logical qubit at physical error rate \(0.1\%\) for logical CNOT error rates \(10^{-10}\) and \(10^{-24}\), respectively.
• \([[6,2,2]]\) \(C_6\) code— Concatenating quantum Hamming codes on top of the \([[4,2,2]]\) and \(C_6\) codes yields fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [4]. In the optimized protocol of Ref. [4], a level-five \(C_4/C_6\) code underlies concatenated quantum Hamming codes \(\mathcal{Q}_5,\mathcal{Q}_6,\mathcal{Q}_7,\mathcal{Q}_7\), yielding a \(2.5\%\) threshold and space overheads \(162\) and \(373\) physical qubits per logical qubit at physical error rate \(0.1\%\) for logical CNOT error rates \(10^{-10}\) and \(10^{-24}\), respectively.
• Kitaev surface code— Quantum Hamming codes can be concatenated with surface codes [6]. In a unified logical-CNOT comparison under circuit-level depolarizing noise, using the surface code as the underlying code gives a \(0.31\%\) threshold and requires space overhead \(4.5\times 10^3\) at physical error rate \(0.1\%\) to achieve logical CNOT error rate \(10^{-24}\), compared to \(3.7\times 10^2\) for the optimized \(C_4/C_6\)/Hamming construction [4].
• Quantum data-syndrome (QDS) code— Because every stabilizer generator has the same weight \(2^{r-1}\), quantum Hamming codes admit QDS extensions based on good binary syndrome-measurement codes [7].