Binary integer programming for optimizing ebit cost in distributed quantum circuits with fixed module allocation

TL;DR

BIP post-processing reduces ebit cost by up to 20\% for random circuits and by more than an order of magnitude for some arithmetic circuits and while the method incurs offline classical overhead, it is amortized when circuits are executed repeatedly.

Abstract

Modular and networked quantum architectures can scale beyond the qubit count of a single device, but executing a circuit across modules requires implementing non-local two-qubit gates using shared entanglement (ebits) and classical communication, making ebit cost a central resource in distributed execution. The resulting distributed quantum circuit (DQC) problem is combinatorial, motivating prior heuristic approaches such as hypergraph partitioning. In this work, we decouple module allocation from distribution. For a fixed module allocation (i.e., assignment of each qubit to a specific Quantum Processing Unit), we formulate the remaining distribution layer as an exact binary integer programming (BIP). This yields solver-optimal distributions for the fixed-allocation subproblem and can be used as a post-processing step on top of any existing allocation method. We derive compact BIP formulations for four or more modules and a tighter specialization for three modules. Across a diverse benchmark suite, BIP post-processing reduces ebit cost by up to 20\% for random circuits and by more than an order of magnitude for some arithmetic circuits. While the method incurs offline classical overhead, it is amortized when circuits are executed repeatedly.

Figures (22)

• Figure 1: Example network of five QPUs. Every pair of modules shares entanglement resources to facilitate non-local gate operations. This distributed architecture enables large-scale quantum circuits through a combination of local quantum operations and inter-module classical communication.
• Figure 2: Implementation of a non-local gate. Module $A$ contains qubits $(a,e)$ and Module $B$ contains qubits $(e^\prime,b)$. The controlled-$U$ gate $CU_{a\to b}$ with control $a$ and target $b$ can be realized using only local operations and classical communication at the cost of a single ebit $|\Phi^+\rangle_{ee^\prime}$ shared between $A$ and $B$.
• Figure 3: (a) Example distributed circuit for $k=3$ modules. Wires with the same color belong to the same module. The labels $1,1,2,2,3,3$ on the left are the module indices of the six logical qubits (each module contains two qubits). White squares represent single-qubit gates and the symbols $\alpha,\beta,\gamma,\delta$ denote two-qubit gates (with repeated $\delta$-blocks). Downward arrows indicate migrations (here, to module 3). (b) Hypergraph representation of the same instance, with qubit vertices $q_1,\ldots,q_6$ and gate vertices $\alpha,\beta,\gamma,\delta$. Qubit-vertex colors indicate their home modules, whereas gate-vertex colors indicate the modules where the corresponding gates are executed.
• Figure 4: Illustration of why $k=3$ admits a reduced formulation (Lemma \ref{['lemma:k_3_prune_migrations']}) while $k\ge4$ does not. Gray boxes show module indices (each module contains a single qubit), and arrows denote migrations. (a) $k=3$: Removing the red migration makes the red gate uncovered, but it can still be covered by the single blue migration because the blue migration (which is an element of $M^{(1)}$) moves the same qubit to the other endpoint's home module. This reflects that migrations outside $M^{(1)}$ can be replaced without increasing cost. (b) $k=4$: The circuit includes an additional set of gates, and removing the red migration leaves the red gates uncovered in a way that cannot be repaired by any single migration. This contrast explains why we can restrict migrations to $M^{(1)}$ for $k=3$, but not for $k\ge4$.
• Figure 5: (a) Distribution found by a hypergraph partitioner and (b) BIP post-processing. Numbers denote qubit indices and wires of the same color are assigned to the same module. Arrows indicate migrations. The ebit cost drops from $4$ in (a) to $3$ in (b). The fact that the hypergraph partitioner is suboptimal even on this tiny instance motivates our BIP post-processing step.

Theorems & Definitions (14)

• Definition 1: Migration g2021efficient
• Definition 2: Home coverage g2021efficient
• Definition 3: Joint coverage g2021efficient
• proof
• Lemma 1
• proof
• proof
• Remark 1
• Definition 4
• Remark 2