Quantified convergence of general homodyne measurements with applications to continuous variable quantum computing

Abstract

In arXiv:2503.00188 we introduced broadband pulsed (BBP) homodyne measurements as a generalization of standard pulsed homodyne quadrature measurements. BBP can take advantage of detectors such as calorimeters that have the potential for high efficiency over a broad spectral range. BBP homodyne retains the advantages of standard pulsed homodyne, enabling measurement of arbitrary quadratures in the limit of large amplitude local oscillators (LO). Here we quantify the convergence of standard and BBP homodyne quadrature measurements to those of the quadrature of interest. We obtain lower bounds on the fidelity of the post-measurement classical-quantum state of outcomes and unmeasured modes, and the fidelity of the states obtained after applying operations conditional on measurement outcomes. The bounds depend on the LO amplitude and the moments of number operators. We demonstrate the practical relevance of these bounds by evaluating them for standard pulsed homodyne used for estimating values of the characteristic function of the Wigner distribution, expectations of moments, for quantum teleportation and for continuous variable error correction with GKP codes.

Figures (5)

• Figure 1: Multi-mode BBP homodyne configuration. The signal modes enter from the left on modes ${\bm{a}}$. The LO modes enter from the bottom on modes ${\bm{b}}$. For mathematical convenience, we represent the LO modes as initially in vacuum, with the LO coherent state prepared by the displacement operator $\hat{D}_{R\bm{\beta}}$ . Modes ${\bm{a}}$ and ${\bm{b}}$ are combined on a balanced beam splitter (BS). The beamsplitter's outgoing modes are ${\bm{c}}$ and ${\bm{d}}$. They are measured with detectors such as calorimeters. The observables associated with the detectors are of the form $\hat{E}_{\bm{f}}=\sum_{k}\omega_{k}\hat{f}_{k}^{\dagger} \hat{f}_{k}$ with $\bm{f}=\bm{c}$ or $\bm{d}$, where $\omega_{k}$ are mode energies or other known positive weights. The homodyne measurement result is determined by subtracting one detector's output from the other and rescaling the result by a factor of $1/R$. The corresponding effective measurement operator is shown.
• Figure 2: An explicit circuit diagram for the measurement process with effective apparatus coupling $\hat{M}_{\delta}=e^{-i\hat{q}_{\delta}\otimes \hat{s}_{M}}$. Here we show one way of representing $\hat{M}_{\delta}$ explicitly in terms of the elements depicted in the schematic for BBP homodyne of Fig. \ref{['fig:bbp_homodyne']}. The operator $\hat{E}_{c}-\hat{E}_{d}$ in the diagram denotes the Schrödinger picture operator and is the physical energy difference between the two mode lines at the point where it is used. The coupling to the apparatus in this unitary measurement model is directly in terms of the scaled energy difference operator shown in Fig. \ref{['fig:bbp_homodyne']}, which is converted to $\hat{q}_{\delta}$ as defined in Eq. \ref{['eq:qdelta']} by conjugating with the beamsplitter and displacement. The reversal of the beamsplitter and the displacement is used to define $\hat{M}_{\delta}$ in the unitary measurement model. Since they act on subsequently ignored, destroyed or discarded systems they and any other subsequent action on these systems do not matter for the fidelity analysis. In experiments, the actual measurement may be performed as described in Fig. \ref{['fig:bbp_homodyne']}. The system $B$ includes all other relevant systems including any purifying systems.
• Figure 3: Circuit diagrams for the main states in the distance bounding chain of Eq. \ref{['eq:condchain']}.
• Figure 4: GKP teleported error correction. The top line carries mode $a$ which contains the incoming GKP qubit, the bottom line carries mode $c$ which contains the outgoing GKP qubit. The mode operators for the incoming modes and for the modes after coupling are shown with their relationships. The couplings are denoted by CNOT symbolic gates. These are active linear optical. For example, the one that couples mode $c$ to mode $b$ commutes with $\hat{q}_{c}$ and $\hat{p}_{b}$ and the quadratures $\hat{q}_{b}"$ and $\hat{p}_{c}"=\hat{p}_{c}'$ are given by $\hat{q}_{b}+\hat{q}_{c}$ and $\hat{p}_{c}-\hat{p}_{b}$ in terms of the incoming quadratures. The diagram is based on Fig. 10 of Ref. grimsmo2021quantum.
• Figure 5: Entanglement infidelity as a function of displacement noise strength for square GKP qubits. The displacement noise is Gaussian and isotropic with variance $\sigma_{\text{noise}}^2$. Each series corresponds to a GKP code with with variance $\sigma_{\text{gkp}}^2$, average photon number $\bar{n}$ of the code space projector, or the given squeezing in decibels (dB). According to Ref. aghaee2025scaling, for general GKP fault tolerance, a squeezing safely above $9.75\;\textrm{dB}$ is sufficient. See Fig. S15 of the supplementary information of the reference, which shows that the logical error rate at the threshold is greater than $10\;\textrm{\%}$ and error rates for reasonable coding distances do not drop below $.1\;\textrm{\%}$ for squeezings less than $10\;\textrm{dB}$. Ref.vahlbruch2024detection claims $15\;\textrm{dB}$ optical squeezing. We also include the TRIV curve for the trivial qubit code spanned by the oscillator ground $\ket{0}$ and first excited $\ket{1}$ states. The solid lines correspond to the numerically optimized recovery while the dashed lines correspond to an analytic approximation for teleported error correction circuit in Fig. \ref{['fig:gkptelerr']}. For the analytic approximation, $\sigma_{\text{gkp}}^{2}$ is treated as an effective noise on ideal GKP qubits. Because each of the two ancillas has the same effective noise, by the error propagation analysis of Sect. \ref{['sect:cvec']}, this corresponds to an effective noise variance of $3\sigma_{\text{gkp}}^{2}$ on the incoming GKP qubit with ideal ancillas. For the curves shown, noise with variance $\sigma_{\text{noise}}^{2}$ is added to the incoming GKP qubit only. For the numerically optimized recovery, there are no ancillas. The incoming GKP qubit is the same as for the analytic approximation with squeezing corresponding to $\sigma_{\text{gkp}}$. For comparing to the analytic approximation, we optimized the recovery for an added displacement noise with variance $\sigma_{\text{opt}}^{2}=2\sigma_{\text{gkp}}^{2}+\sigma_{\text{noise}}^{2}$. The straight solid lines going off to the left connect to the infidelity of the optimal recovery for a value of $\sigma_{\text{opt}}^{2}$ that is less than $2\sigma_{\text{gkp}}^{2}$. Their horizontal coordinates correspond to a negative values $-x$ of $\sigma_{\text{noise}}^{2}$, which for illustration purposes were identified with the unplotted locations of $-\sqrt{x}$ on the horizontal axis

Theorems & Definitions (26)

• Theorem 5.1
• proof
• Theorem 5.2
• proof
• Proposition 5.3
• proof
• Proposition 5.4
• proof
• Proposition 6.1
• proof