Optimal logical Bell measurements on stabilizer codes with linear optics

TL;DR

The paper addresses the inefficiency of linear-optics Bell measurements when reading stabilizer-encoded logical qubits. It develops a universal, group-theoretic framework showing that any two-code logical BM with probabilistic physical BMs can be bounded by the likelihood of at least one successful physical BM, yielding the no-loss bound $1 - (1 - \mathbb{P}_B)^{\min(n_1,n_2)}$. It then provides sufficient conditions for optimal, feedforward-based BM schemes and demonstrates explicit optimal constructions for several codes (quantum parity, five-qubit, standard and rotated planar surface codes, tree, Steane), many achieving the bound, with an optimized static scheme for the rotated planar code. The results significantly improve logical BM success probabilities in lossless optical settings and offer a rigorous framework for scalable, fault-tolerant photonic quantum technologies, while outlining future work on photon loss and broader code families.

Abstract

Bell measurements (BMs) are ubiquitous in quantum information and technology. They are basic elements for quantum commmunication, computation, and error correction. In particular, when performed on logical qubits encoded in physical photonic qubits, they allow for a read-out of stabilizer syndrome information to enhance loss tolerance in qubit-state transmission and fusion. However, even in an ideal setting without photon loss, BMs cannot be done perfectly based on the simplest experimental toolbox of linear optics. Here we demonstrate that any logical BM on stabilizer codes can always be mapped onto a single physical BM perfomed on any qubit pair from the two codes. As a necessary condition for the success of a logical BM, this provides a general upper bound on its success probability, especially ruling out the possibility that the stabilizer information obtainable from only partially succeeding, physical linear-optics BMs could be combined into the full logical stabilizer information. We formulate sufficient criteria to find schemes for which a single successful BM on the physical level will always allow to obtain the full logical information by suitably adapting the subsequent physical measurements. Our approach based on stabilizer group theory is generally applicable to any stabilizer code, which we demonstrate for quantum parity, five-qubit, standard and rotated planar surface, tree, and seven-qubit Steane codes. Our schemes attain the general upper bound for all these codes, while this bound had previously only been reached for the quantum parity code.

Figures (32)

• Figure 1: This figure illustrates a crucial aspect of the proof for Thm. \ref{['thm:bound']}. Any pair of logical operators $\overline{XX}$ and $\overline{ZZ}$ anticommutes in two qubits $\lambda_1$ and $\lambda_2$. In the proof we argue that any code stabilizer $\gamma$ with support on $\lambda_1$ and $\lambda_2$ must also anticommute with one of the logical operators in two other qubits $\lambda_1^\prime$ and $\lambda_2^\prime$.
• Figure 2: Logical BM scheme for QPC($2,2$). Each gray box represents a single logical qubit encoded in four physical qubits, depicted as circles. Red and blue indicates $XX$-BMs and $ZZ$-BMs, respectively. The qubits filled with both red and blue indicate a successful physical BM.
• Figure 3: Logical BM scheme for QPC($2,2$) in the single-code picture. Each gray box represents a single logical qubit encoded in four physical qubits, depicted as circles. Red and blue indicates $X$-BMs and $Z$-BMs, respectively. Qubits filled with both red and blue indicate a successful physical BM. The red and blue lines illustrate the measured logical operators $\overline{X}$ and $\overline{Z}$, respectively.
• Figure 4: Examples of supports of a logical $\overline{X}$ (left) and $\overline{Z}$ (right) operators for QPC. Qubits on which the logical operators act with $X$, $Z$ or the identitiy are shown in red, blue, and white, respectively. The red and blue strings represent $\overline{X}$ and $\overline{Z}$, respectively. The code dimensions $(r, m)$ are shown in the left example.
• Figure 5: Example for the measurement scheme for QPC($5,5$). In this example, the first two rows were measured without any successful BM. A successful BM occurs on the second qubit of the third row. The scheme is then completed by performing $X$-BMs on the remaining qubits of that row and $Z$-BMs on one qubit from each of the remaining rows. In this figure, we have chosen the last qubit of each remaining row as an example. The red and blue strings represent the measured $\overline{X}$ and $\overline{Z}$, respectively.

Theorems & Definitions (56)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 1
• proof