Description
Two-mode bosonic code encoding a logical qubit in Fock states with one excitation. The logical-zero state is represented by \(|10\rangle\), while the logical-one state is represented by \(|01\rangle\). This encoding is often realized in temporal or spatial modes, corresponding to a time-bin or frequency-bin encoding. Two different types of photon polarization can also be used.
Protection
This is an error-detecting code against one photon loss event; it is often used in photonic quantum devices because of its ease of realization. A single loss event can be detected because, after the loss occurs, the output state \(|00\rangle\) is orthogonal to the codespace. Recovery is not possible, so a successful run of a quantum circuit is conditioned on not losing a photon during the circuit.
For Deutsch''s problem specifically, this code protects against errors resulting in states that have the correct photon number, but in the wrong modes [9].
Encoding
Optimal control pulses [10]Gates
General gates are performed using two-body Hamiltonian rotations [8].Bosonic gates include beamsplitters [11] and Kerr nonlinearities. In particular, a cross Kerr rotation at angle \(\pi\) induces a \(CZ\) gate. Universal quantum computing can be achieved using the KLM protocol [12] with only linear optical elements and photon detectors.Dynamical-decoupling protocols [8,13].A probabilistic CZ gate via a non-linear sign-shift gate, which transforms the Fock states \( \alpha|0\rangle+\beta|1\rangle+\gamma|2\rangle\) into \(\alpha|0\rangle+\beta|1\rangle-\gamma|2\rangle \), followed by measurement [14].Error-detecting \(CCZ\) and \(cSWAP\) gates using three-level ancilla [15].Cavity-assisted bias-preserving CNOT gate [16].Fault Tolerance
Dual-rail qubits can be used to convert leakage and AD noise into erasure noise [17,18].Realizations
The dual-rail code is ubiquitous in linear-optical quantum devices and is behind the KLM protocol, one of the first proposals for fault-tolerant computation. See reviews [19–21] for more details.Superconducting circuit devices: Gates have been demonstrated in the Schoelkopf group at Yale University [11]. Error detection has been demonstrated in 3D cavities in the Devoret group at Yale University [10] and Amazon Web Services [22] using transmon qubits, following earlier theoretical proposals [18,23]. GHZ and Bell states as well as universal gates have been implemented on a four-qubit dual-rail code by the Yu group [24]. Logical readout in 3D cavities has been demonstrated by Quantum Circuits Inc. [25]. Cavity-assisted bias-preserving CNOT gate has been demonstrated [16].Photonic platforms: state preparation and measurement fidelity of \(99.98\%\) in the C telecom band by PsiQuantum [26].Cousins
- Concatenated bosonic code— The KLM protocol, one of the first protocols for fault-tolerant quantum computation, utilizes concatenations of the dual-rail code with a stabilizer code such as the Steane code [12,27,28]. Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [29] that protects against \(d-1\) AD errors [30]. Concatenating the outer dual-rail code with an inner single-mode bosonic code yields several gates that are independent of the inner code [31].
- \([[7,1,3]]\) Steane code— The KLM protocol, one of the first protocols for fault-tolerant quantum computation, utilizes concatenations of the dual-rail code with a stabilizer code such as the Steane code [12,27,28].
- Single-mode bosonic code— Concatenating the outer dual-rail code with an inner single-mode bosonic code yields several gates that are independent of the inner code [31].
- Amplitude-damping (AD) code— Dual-rail qubits can be used to convert leakage and AD noise into erasure noise [17,18]. Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [29] that protects against \(d-1\) AD errors [30].
- Quantum parity code (QPC)— An \([[8,1,2]]\) QPC correcting a single AD error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code [29,32,33]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [30].
- \([[4,2,2]]\) Four-qubit code— An \([[8,1,2]]\) QPC correcting a single AD error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code [29,32,33]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [30].
- Cluster-state code— The KLM protocol can be combined with cluster states in various ways to yield MBQC protocols [34–36]; see review [21].
- Quantum repetition code— The dual-rail code is an error space of the quantum repetition code for \(n=2\) and is stabilized by \(-ZZ\).
- Fusion-based quantum computing (FBQC) code— FBQC resource states are concatenated with dual-rail codes to increase loss detection.
- XYZ\(^2\) hexagonal stabilizer code— The XYZ\(^2\) hexagonal stabilizer code can be viewed as a concatenation of the \(YZZY\) surface code with one of the possible \([[2,1]]\) repetition codes, with the case of the bit-flip repetition code yielding a concatenation of the surface code with the dual-rail code [37].
Primary Hierarchy
Parents
Two-mode binomial codeFock-state bosonic CE Amplitude-damping (AD) Hamiltonian-based QECC AQECC Quantum
The two-mode binomial code for \(S=N=0\) reduces to the dual-rail code.
Dual-rail quantum code
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Page edit log
- Yinchen Liu (2024-03-15) — most recent
- Esha Swaroop (2024-03-14)
- Victor V. Albert (2021-12-18)
- Dhruv Devulapalli (2021-12-17)
- Aniket Maiti (2024-02-08)
Cite as:
“Dual-rail quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/dual_rail