Characterizing Space-Constrained Implementability of Quantum Instruments via Signaling Conditions

TL;DR

The paper addresses how many qubits are needed to implement a general quantum instrument under space constraints, especially when mid-circuit measurements and delayed input preparation are allowed. It develops a signaling-causal framework that yields upper and lower bounds on qubit requirements via instrument composability and a novel outcome no-signaling condition, and then applies these results to entanglement-distillation protocols based on stabilizer codes. In stabilizer-code settings, the derived bounds are tight, enabling exact qubit counts: 3 qubits for the $[[9,1,3]]$ protocol and 4 qubits for the $[[5,1,3]]$ protocol (with related results for the $[[7,1,3]]$ code). These findings provide concrete resource estimates for space-constrained quantum operations and have practical implications for distributed quantum processing and circuit compilation.

Abstract

Scaling up the number of qubits available on quantum processors remains technically demanding even in the long term; it is therefore crucial to clarify the number of qubits required to implement a given quantum operation. For the most general class of quantum operations, known as quantum instruments, the qubit requirements are not well understood, especially when mid-circuit measurements and delayed input preparation are permitted. In this work, we characterize lower and upper bounds on the number of qubits required to implement a given quantum instrument in terms of the causal structure of the instrument. We further apply our results to entanglement distillation protocols based on stabilizer codes and show that, in these cases, the lower and upper bounds coincide, so the optimal qubit requirement is determined. In particular, we compute that the optimal number of qubits is 3 for the $[[9,1,3]]$-code-based protocol and 4 for the $[[5,1,3]]$-code-based protocol.

Figures (23)

• Figure 1: Two examples of reducing the number of qubits required to execute quantum instruments. Here, $\Lambda := \{\Lambda_{(i_1, i_2)}\}_{(i_1,i_2)}$ and $\Gamma := \{\Gamma_{(i_1, i_2)}\}_{(i_1,i_2)}$ are quantum instruments, and $U_1$ and $U_2$ represent two-qubit unitary operations. In each case, the quantum instrument shown on the left-hand side can be implemented with two qubits, as depicted on the right. (a) The quantum instrument $\Lambda$ has two input qubits $\mathrm{A}_1$ and $\mathrm{A}_2$, and employs one auxiliary qubit in its Stinespring dilation. The mid-circuit measurement is performed for outcome $i_1$, and the measured qubit is reused for the initialization of the auxiliary qubit. (b) The quantum instrument $\Gamma$ has three input qubits $\mathrm{A}_1$, $\mathrm{A}_2$, and $\mathrm{A}_3$. The input state of $\mathrm{A}_3$ is prepared after the circuit execution starts, and loaded after the mid-circuit measurement for outcome $i_1$.
• Figure 2: Unitary operation.
• Figure 3: Computational basis measurement.
• Figure 4: State reset.
• Figure 5: Classical processing.

Theorems & Definitions (50)

• Definition 1: $m$-qubit Implementable Instruments (without Delayed Inputs)
• Remark 1
• Remark 2
• Definition 2: $m$-Qubit Implementable Instruments (with Delayed Inputs)
• Remark 3
• proof : The proof sketch of \ref{['rem:stairs_wdi']}
• Remark 4
• Lemma 3
• proof
• Definition 4: Composability of Quantum Instruments (Lepp2021)