Abstract

Estimating and reducing the overhead of fault tolerance (FT) schemes is a crucial step toward realizing scalable quantum computers. Of particular interest are schemes based on two-dimensional (2D) topological codes such as the surface and color codes which have high thresholds but lack a natural implementation of a non-Clifford gate. In this work, we directly compare two leading FT implementations of the T gate in 2D color codes under circuit noise across a wide range of parameters in regimes of practical interest. We report that implementing the T gate via code switching to a 3D color code does not offer substantial savings over state distillation in terms of either space or space-time overhead. We find a circuit noise threshold of 0.07(1)% for the T gate via code switching, almost an order of magnitude below that achievable by state distillation in the same setting. To arrive at these results, we provide and simulate an optimized code switching procedure, and bound the effect of various conceivable improvements. Many intermediate results in our analysis may be of independent interest. For example, we optimize the 2D color code for circuit noise yielding its largest threshold to date 0.37(1)%, and adapt and optimize the restriction decoder finding a threshold of 0.80(5)% for the 3D color code with perfect measurements under Z noise. Our work provides a much-needed direct comparison of the overhead of state distillation and code switching, and sheds light on the choice of future FT schemes and hardware designs.

Highlights

  • MATERIAL we review some relatively standard but important background material that we will refer to throughout the paper

  • We find asymptotic expressions that support our finding that state distillation requires lower overhead than code switching

  • V is the culmination of our work, where building upon results from the previous sections we provide a simplified recipe for code switching, detailing each step, and specifying important optimizations

Read more

Summary

BACKGROUND

We review some relatively standard but important background material that we will refer to throughout the paper.

Noise and simulation
Basics of 2D and 3D color codes
Fault-tolerant computation with 2D color codes
State distillation
Projection decoder with boundaries
Noisy-syndrome projection decoder with boundaries
Optimizing stabilizer extraction and circuit-noise analysis
Modeling noise in logical operations
Creating the T state via state distillation in three steps
T-state initialization
Expansion and movement of patches
State-distillation overhead
INSIGHTS INTO 3D COLOR CODES
A simple way to switch between 2D and 3D color codes
Physics of the gauge flux in 3D color codes
Restriction decoder for 3D color codes with boundaries
Creating the T state via code switching in six steps
Preparing the Bell state in 2D codes
Preparing the 3D interior
Measuring gauge operators
Gauge fixing
Applying T and measuring the 3D code’s data qubits
Decoding Z errors in the 3D color code
Code-switching overhead
Findings
DISCUSSION
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call