Description
A squeezed Fock-state code constructed from a coherent superposition of \(m\) squeezed vacuum states, each squeezed along evenly-spaced axes in phase space.
For an even integer \(m > 0\) (the number of “legs”) and squeezing strength \(r\), the two logical codewords are defined as \begin{align} |0_L\rangle & \propto \sum_{j=0}^{m-1} S\left(r, \frac{\pi j}{m}\right) \ket{\vac}, \tag*{(1)}\\ |1_L\rangle & \propto \sum_{j=0}^{m-1} (-1)^{j} S\left(r, \frac{\pi j}{m}\right) \ket{\vac}, \tag*{(2)}\end{align} where \(S(r,\theta) \equiv S(r e^{i\phi(\theta)})\) is the squeezing operator with \(\phi(\theta) = 2\theta + \pi \pmod{2\pi}\), defined as \begin{equation} S(\zeta) = \exp\!\left[\frac{1}{2}\left(\zeta^{*} a^{2}-\zeta\, a^{\dagger 2}\right)\right], \tag*{(3)}\end{equation} and where \(\zeta = r e^{i\phi}\) and \(r\) is the squeezing strength. This operator elongates the vacuum state along direction \(\theta\) in phase space.
Protection
In Fock space, the logical states occupy photon numbers \(n \equiv 2k \pmod{2m}\) for \(k \in \{0, m/2\}\), yielding natural number distributions that are interleaved by \(\Delta n = m\). The code distance against single-photon loss is \(d = m\), where \(m\) is the number of legs (squeezed vacuum states in superposition). The code provides protection against both photon loss and dephasing noise, with a fundamental trade-off: increasing \(m\) improves loss tolerance at the cost of higher dephasing sensitivity.Encoding
Probabilistic preparation for \(m=2\): The simplest circuit uses a Hadamard-Controlled-Squeezing-Hadamard (\(H\)-\(CS\)-\(H\)) sequence on a single ancilla qubit coupled to a bosonic mode initially in vacuum. The controlled squeezing gate is defined as \begin{equation} CS(r;\theta_0,\theta_1) = \ket{0}\!\bra{0}\otimes S(r,\theta_0) + \ket{1}\!\bra{1}\otimes S(r,\theta_1). \tag*{(4)}\end{equation} After applying \(H\)-\(CS(r;0,\pi/2)\)-\(H\) and measuring the ancilla in the \(Z\) basis, the bosonic mode collapses to either \(\ket{0_L}\) or \(\ket{1_L}\) with probabilities \begin{equation} \text{prob}(L \mid r) = \frac{1}{2} + \frac{(-1)^L}{2 \cosh r \sqrt{\tanh^2 r + 1}}, \tag*{(5)}\end{equation} where \(L \in \{0, 1\}\). Post-selection on the measurement outcome yields the desired logical state. Deterministic preparation for general \(m\): For \(m > 2\), the codes can be prepared using sequences of conditional rotations \(CR(\theta)\) that rotate the bosonic mode in phase space by angle \(\theta\) conditioned on the qubit state, combined with logical-\(X\) gates that flip between the computational basis states. The circuit involves creating superpositions with single-qubit gates and applying conditional operations in a recursive manner. With full single-qubit control and controlled-squeezing operations, these circuits provide universal control over the joint qubit-oscillator system. Recursive construction: Higher-\(m\) codes can be generated by feeding back the output of lower-\(m\) code preparation circuits, applying additional conditional rotations, and selecting appropriate measurement outcomes. This allows systematic construction of the entire code family.Gates
Universal computation requires multiple bosonic modes and entangling gates between them, such as a controllable cross-Kerr interaction \(a^\dagger a \otimes a^\dagger a\) that enables generation of the entangling \(CZ\) gate.Logical operations within a single mode can be performed using Gaussian operations (displacement, rotation, squeezing) combined with conditional control from an ancilla qubit.Decoding
The interleaved photon-number structure (\(n \equiv 2k \pmod{2m}\)) enables photon-number-resolving measurements to identify single-photon loss events, which can then be corrected.Cousin
- Number-phase code— Squeezed-vacuum codes are approximate number-phase codes with Fock-space support \(n \equiv 2k \pmod{2m}\), approaching ideal number-phase codes as squeezing strength \(r \to \infty\) [2].
Primary Hierarchy
References
- [1]
- M. K. Hope, J. Lidal, and F. Massel, “Preparation of conditionally-squeezed states in qubit-oscillator systems”, (2025) arXiv:2504.01664
- [2]
- N. Gutman, E. Blumenthal, S. Hacohen-Gourgy, A. Orda, and I. Kaminer, “Squeezed-vacuum bosonic codes”, (2025) arXiv:2511.06108
- [3]
- A. L. Grimsmo, J. Combes, and B. Q. Baragiola, “Quantum Computing with Rotation-Symmetric Bosonic Codes”, Physical Review X 10, (2020) arXiv:1901.08071 DOI
Page edit log
- Victor V. Albert (2025-11-18) — most recent
- Nir Gutman (2025-11-18)
Cite as:
“Squeezed-vacuum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/squeezed_vacuum