[Paper Review] Cleaner magic states with hook injection

Craig Gidney|arXiv (Cornell University)|Feb 23, 2023

Advanced Data Storage Technologies8 citations

TL;DR

The paper introduces hook injection, an intentional hook-error mechanism to inject magic states into surface codes, limited to XY or YZ planes, and shows via Monte Carlo that it achieves lower injection error rates and smaller spacetime costs than prior methods.

ABSTRACT

In this paper, I show how an intentional hook error mechanism can be used as a control knob for injecting magic states into surface codes. The limitation, and benefit, of this approach is that it can only inject states in the XY or YZ plane of the Bloch sphere. This increases fidelity, because perturbations out of the target plane can be detected as errors. I use Monte Carlo sampling to show that this technique outperforms previous injection techniques, achieving lower error rates at smaller spacetime cost under digitized circuit noise.

Motivation & Objective

• Motivate the need for high-fidelity magic state injection to reduce overall distillation costs in fault-tolerant quantum computation.
• Propose hook injection as a programmable, plane-constrained method to inject magic states into a distance-2 surface code patch.
• Demonstrate improvements over Li15 and Sin+22 injection methods using simulations and noise models.
• Assess tradeoffs between injection error rate, patch size, and postselection rounds to guide practical deployment.

Proposed method

• Use an intentional hook error during surface code stabilizer measurement to rotate the logical observable, injecting a state in the XY (or YZ) plane.
• Grow a distance-2 patch to an intermediate distance d_inject while measuring stabilizers, with a specific qubit/axis assignment to minimize distance-1 and distance-2 error pathways.
• Restrict the initialization basis across the patch and remove no-op CZs to reduce error mechanisms and circuit depth.
• Model injection fidelity with digitized Pauli noise and a superconducting-inspired noise model (SI1000), and count undetectable distance-1 and distance-2 errors to bound injection error rates.
• Benchmark hook injection against Li15 and Sin+22 variants via Pareto-front analyses of injection error rate versus expected spacetime cost per successful injection.
• Explore variants including pregrown hook injection and parameter sweeps over d_inject and r_inject to map performance under different discard rates.

Figure 1: Defect diagram of the injection process. Red surfaces are X boundaries, blue surfaces are Z boundaries, and magenta bars are twist defects. Attempts are made in parallel at a low distance $d_{\text{inject}}$ , until an attempt succeeds and expands to the target distance $d$ , as in [ Li15

Experimental results

Research questions

• RQ1 How does intentional hook error control affect the set of injectable magic states within the XY or YZ plane?
• RQ2 Does hook injection achieve lower injection error rates and lower spacetime costs than Li15 and Sin+22 under digitized noise?
• RQ3 What are the tradeoffs between patch size (d_inject), postselection rounds (r_inject), discard rate, and injection success under realistic noise models?
• RQ4 How do different injection configurations perform across ranges of noise strength (p) and what are practical guidance for factory integration?

Key findings

• Hook injection achieves lower injection error rates than prior methods at comparable or smaller spacetime costs.
• Under the SI1000 noise model at p=0.1%, d_inject=5 yields injection error rate below 0.1% with discard rate below 50%; d_inject=7 yields injection error rate below 0.06% with discard rate below 75%.
• Distance-1 and distance-2 error mechanisms place a hard lower bound on injection error rates, e.g., approximately 7p/30 for |i> and 5p/30 for |+> under some conditions.
• The injection cost frontiers (Pareto) show hook injection outperforms Li15 and ZZ injection across tested parameters.
• Pregrown hook injection demonstrates potential for good error rates with postselection on subregions, suggesting room for improved tradeoffs.
• The results indicate hook injection changes prior optimistic estimates of distillation cost by accounting for full circuit noise, rather than merely interaction-dominated models.

Figure 2: Detector slice diagrams of hook injection with $d_{\text{inject}}=5$ , $r_{\text{inject}}=2$ , $d=7$ . Layers advance in reading order, from left to right then top to bottom. Measurements outlined in teal are predictable from previous measurements, and are used to detect errors. If any of