Lattice surgery with Bell measurements: Modular fault-tolerant quantum computation at low entanglement cost

TL;DR

The paper tackles inter-module fault-tolerance by proposing a lattice-surgery protocol based on Bell measurements for rotated surface codes in modular quantum architectures. This approach confines interface noise and reduces the per-round entanglement cost to $d$ ebits, improving resource efficiency over prior $2d-1$ Bell-pair schemes. Circuit-level depolarizing-noise simulations show stronger logical-error suppression per entanglement unit, with typical ~40% entanglement savings to achieve a fixed logical error rate, and robust performance across a range of link-noise levels. The work also introduces alternating syndrome-measurement sequences to mitigate distance-reducing hook errors and discusses broader applicability to distributed quantum circuits beyond the surface code.

Abstract

Modular architectures are a promising approach to scaling quantum computers to fault tolerance. Small, low-noise quantum processors connected through relatively noisy quantum links are capable of fault-tolerant operation as long as the noise can be confined to the interface. Finding protocols that implement the quantum links between modules as efficiently as possible is essential because inter-module entanglement is challenging to produce at a similar rate and fidelity as local entanglement. We introduce a protocol for lattice surgery on surface codes in which all non-local operations are Bell measurements. The protocol simultaneously confines the link noise and requires only half as many module-crossing gates as previously proposed protocols. To mitigate distance-reducing hook errors, we introduce a strategy of alternating the gate sequence between rounds of syndrome measurement, which prevents multiple hooks from simultaneously aligning with a logical operator in the code. We evaluate our protocol's performance when two logical qubits on separate modules are prepared in a logical Bell state. Circuit-level simulations under depolarizing noise show that the logical error suppression for a given entanglement rate between modules is consistently stronger compared to the best-performing alternative protocols for a wide range of link noise, with a typical 40% entanglement resource saving for a constant logical error rate. Our approach to protocol design is applicable to any quantum circuit that must be divided across processor modules and can therefore guide development of resource-efficient modular quantum computation beyond the surface code.

Figures (10)

• Figure 1: Modular quantum-computing architectures and lattice surgery on surface codes. (a) An example modular architecture for a superconducting quantum computer with four quantum processing units (QPUs) connected by direct links (orange) through an interposer chip. Each QPU is also equipped with photonic connections (blue) that support entanglement distribution with itinerant photons to a QPU on another module. (b) Lattice surgery with surface-code logical qubits on separate modules connected by quantum links. Dark (light) squares represent X(Z)-stabilizers of the surface code. When patches are merged, physical qubits on separate modules interact by consuming entanglement that must be established between modules over the quantum links. (c) The rate of entanglement consumption depends on the protocol that is used for the surgery. Previous protocols have required $2d-1$ Bell pairs per round of stabilizer measurement in a code of distance $d$. Each Bell pair has one ebit of entanglement bennett_concentrating_1996. We introduce a protocol for lattice surgery based on Bell measurements, which only requires $d$ ebits per round of syndrome measurement.
• Figure 2: ZX diagrams for detection of phase flips in the surface code. (a) The simplest ZX diagram in which all spiders are connected to spiders of the opposite type. (b) At the interface between two modules, we split spiders in the diagram so that the number of legs crossing the interface is four. We then insert two-legged spiders of opposite type on each side of the interface to illustrate that links between the two sides of the interface can be implemented with CX gates as part of a Bell measurement. These links are highlighted in orange for clarity. (c) The ZX diagrams can be further expanded to give intuition about how they are implemented with quantum circuits on physical qubits. On the left is an expansion that produces the quantum circuit for measuring surface-code X-stabilizers in the bulk. On the right is the corresponding expansion for measuring the same stabilizer at the interface. (d) Elementary building blocks for ZX diagrams and their interpretations in terms of operations on physical qubits.
• Figure 3: Syndrome-measurement circuits for a rotated surface code. Light (dark) plaquettes represent Z(X)-stabilizers. The syndrome-measurement circuit in the absence of an interface is shown on the square plaquettes. A zig-zag pattern is chosen to avoid $X$ errors propagating vertically and $Z$ errors propagating horizontally. The gate sequence in the Bell-measurement protocol is shown on rectangular plaquettes. Gates crossing the interface are colored red. The gate sequence must be alternated between rounds to avoid distance-reducing hook errors at the interface, as described in the main text.
• Figure 4: Effect of alternating the gate sequence between rounds of syndrome measurement. Local gates and measurements are noiseless and gates crossing the interface are noisy. (a) There are two possible choices for the gate sequence that do not cause hook errors in the bulk, which we label A and B. Applying the same gate sequence for each round of syndrome measurement (AAA/BBB) produces a syndrome graph with parallel diagonal edges. The parallel orientation of the edges represents physical errors aligning with the logical operator of the code to reduce the effective distance of the code at the interface. Alternating the gate sequence between rounds of syndrome measurement (ABA) mitigates this effect by redirecting chains of physical errors. The resulting syndrome graph has a zig-zag pattern of diagonal edges. (b) Simulation of the effective code distance using repeating and alternating gate sequences for syndrome measurement. Repeating the gate sequence every round of syndrome measurement effectively halves the distance of the code. Alternating gate sequences between rounds increases the effective code distance at the interface to $0.9d$, which matches the benchmark protocol.
• Figure 5: Preparation of a Bell state across an interface using a rotated surface code and lattice surgery. Dark boxes represent X-detectors and light boxes represent Z-detectors. At the bottom is a projection of the fault-tolerant channel onto a surface, where the physical layout of ancilla qubits used to extract syndrome information is shown. The interface between the two modules is represented by a dark sheet which bisects the channel.