[Paper Review] Low overhead quantum computation using lattice surgery

Austin G. Fowler, Craig Gidney|arXiv (Cornell University)|Aug 20, 2018

Quantum Computing Algorithms and Architecture68 citations

TL;DR

Lattice surgery in the surface code reduces storage overhead by about 4x and distillation overhead by about 5x, enabling large-scale Clifford+T computation with far fewer physical qubits than defect/braid approaches.

ABSTRACT

When calculating the overhead of a quantum algorithm made fault-tolerant using the surface code, many previous works have used defects and braids for logical qubit storage and state distillation. In this work, we show that lattice surgery reduces the storage overhead by over a factor of 4, and the distillation overhead by nearly a factor of 5, making it possible to run algorithms with $10^8$ T gates using only $3.7 imes 10^5$ physical qubits capable of executing gates with error $p\sim 10^{-3}$. These numbers strongly suggest that defects and braids in the surface code should be deprecated in favor of lattice surgery.

Motivation & Objective

• Develop techniques to implement arbitrary quantum algorithms with lattice surgery.
• Quantify and compare qubit/storage and distillation overhead against defect/braid approaches.
• Provide precise time and space overhead calculations for fault-tolerant quantum computation using lattice surgery.
• Demonstrate practical scalability for large T-gate budgets (e.g., 10^8 T gates) under realistic error rates.

Proposed method

• Adopt rotated logical qubits and lattice-surgery layouts to minimize physical qubits per logical qubit (≈3d^2) versus 12.5d^2 for defect-based qubits.
• Describe procedures for logical initialization, measurement, and movement using stabilizer products.
• Present multi-body X and Z measurements to implement complex logical operators and distillation structures.
• Outline state injection, T-state distillation (including T†) and half-distance rotations, with error propagation p_o = 35 p_i^3.
• Provide gate implementations (S/S†, T/T†, Hadamard, CNOT, CZ) within the lattice-surgery framework.

Experimental results

Research questions

• RQ1 What overhead savings does lattice surgery offer for storage compared with defects and braids in the surface code?
• RQ2 How does lattice surgery impact the distillation overhead and resource requirements for producing high-fidelity T states?
• RQ3 Can a universal gate set (Hadamard, CNOT/CZ, S/S†, T/T†) be efficiently implemented with lattice surgery while maintaining manageable error rates?
• RQ4 What are the estimated physical qubit counts and runtime to execute a Clifford+T circuit with a target T-gate budget (e.g., 10^8 T gates) under realistic error rates?

Key findings

• Lattice surgery reduces logical qubit storage overhead to about 3d^2 physical qubits per logical qubit, a factor of ~4 improvement over simple defect packing.
• Distillation overhead is reduced by nearly a factor of 5 using lattice-surgery-based state preparation and measurement schemes.
• For an algorithm with 10^8 T gates and p ≈ 10^-3, lattice surgery requires ~3.7×10^5 physical qubits and yields comparable runtime to defect/braid baselines.
• Compared to defect-based estimates (e.g., 1.8×10^6 physical qubits for similar tasks), this approach offers substantial qubit savings with similar execution time (roughly hours-scale).
• The paper provides a spreadsheet-based overhead calculation confirming near fivefold qubit savings while maintaining practical runtimes.