Space-Time Optimisations for Early Fault-Tolerant Quantum Computation

TL;DR

This paper addresses resource-efficient compilation for early fault-tolerant quantum computers by jointly optimizing qubit layouts and magic-state routing under strict distillation constraints. It introduces distillation-adaptive layouts and greedy, DAG-informed movement strategies with a Dijkstra-based routing backbone to minimize ancilla usage and distillation bottlenecks. The approach achieves substantial qubit reductions (average ≈53%) with modest time overhead (≈1.2×) and up to ~2× spacetime-volume improvements over some baselines, closely approaching theoretical lower bounds on execution time. The results demonstrate practical, near-optimal space-time tradeoffs for early FTQC, enabling more feasible deployments with tens-to-hundreds of logical qubits.

Abstract

Fault-tolerance is the future of quantum computing, ensuring error-corrected quantum computation that can be used for practical applications. Resource requirements for fault-tolerant quantum computing (FTQC) are daunting, and hence, compilation techniques must be designed to ensure resource efficiency. There is a growing need for compilation strategies tailored to the early FTQC regime, which refers to the first generation of fault-tolerant machines operating under stringent resource constraints of fewer physical qubits and limited distillation capacity. Present-day compilation techniques are largely focused on overprovisioning of routing paths and make liberal assumptions regarding the availability of distillation factories. Our work develops compilation techniques that are tailored to the needs of early FTQC systems, including distillation-adaptive qubit layouts and routing techniques. In particular, we show that simple greedy heuristics are extremely effective for this problem, offering up to 60% reduction in the number of qubits compared to prior works. Our techniques offer results with an average overhead of 1.2X in execution time for a 53% reduction in qubits against the theoretical lower bounds. As the industry develops early FTQC systems with tens to hundreds of logical qubits over the coming years, our work has the potential to be widely useful for optimising program executions.

Figures (17)

• Figure 1: (a) Circuit representation of X and Z syndrome measurements. (b) (Left) The logical qubit has $2d^{2}-1$ physical qubits where d is the code distance (here d=5). The big green circles are the data qubits while the small red circles are the syndrome qubits (red and green faces correspond to the Z and X syndromes, respectively). (Right) Layout of logical qubits in the FTQC system. Orange qubits are the logical data qubits, while the grey region consists of logical ancillas, which are used both for instruction implementation and routing magic states.
• Figure 2: (a) Logical qubit with X and Z edges. (b) Lattice surgery merge and split operations on two logical qubits. (c) CNOT gate decomposed into lattice surgery primitives. It has a control and a target qubit, given by $\ket{c}$ and $\ket{t}$ qubits. The ancilla qubit is $\ket{+}$.(d) CNOT operation implementation on two logical qubits and an ancilla.
• Figure 3: Qubit layouts generated as a function of routing paths. The data qubits are in orange, while the bus qubits are in grey.
• Figure 4: Neighbour-dependent moves inspect neighbouring qubits to determine the appropriate move operation. Before the CNOT operations, qubits labelled 3, 5, 7, and 9 undergo a Hadamard operation simultaneously (a). Following this, the data qubits consult the circuit's directed acyclic graph (DAG) to determine the subsequent move operations. In this instance, the upcoming gates are three CNOTs applied to qubit pairs (3, 4), (5, 6), and (7, 8). To prepare for these interactions, the data qubits move to adjacent positions, enabling diagonal placement (b). The ancilla qubits positioned between each control–target pair are highlighted in neon. The Hadamard operation requires three timesteps, whereas each move operation requires one timestep.
• Figure 5: The starting position is qubit 6 at coordinates (2,1), and the target position is qubit 14 at (5,3). As illustrated in (a), the path looks for minimum disturbance on the grid. Figure (b) shows the corresponding decision tree, where the green cells indicate the optimal movement path. The red cells indicate the rejected cells. The cost of each movement is given on the edge and cells with the minimum cost are chosen.