Magic State Distillation Using CSS Codes for Low Space Overhead
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Solution Overview
Problem
Current quantum computing protocols for magic state distillation have high overhead and require large batches of magic states, with existing methods needing concatenation of small routines to achieve high fidelity, which increases space and computational requirements.
Innovation Solution
The development of a family of distillation protocols that use weakly self-dual Calderbank-Shor-Steane codes to implement control-Swaps and measure properties of magic states, reducing overhead by injecting magic states at various steps in the circuit and leveraging triorthogonal matrices for efficient error detection and correction.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If concatenation of small routines is used to achieve high fidelity, then error suppression is improved, but space overhead and computational requirements increase
Solution Approach 1:
The protocol divides the distillation process into two independent stages: a Clifford stage that prepares intermediate magic states using only Clifford gates, and a non-Clifford stage that completes the distillation to high fidelity. This segmentation allows each stage to be optimized independently, avoiding the need for concatenation while achieving the desired error suppression.
Solution Approach 2:
The protocol introduces intermediate magic states as a mediator between the initial noisy states and the final high-fidelity states. These intermediate states serve as a bridge, allowing the distillation process to proceed in two stages rather than requiring direct concatenation of small routines, thereby reducing space overhead while maintaining reliability.
2Measurement precision
If larger batches of magic states are used, then distillation fidelity is improved, but the number of input magic states and T-gate depth increase
Solution Approach 1:
By segmenting the distillation into two stages, the protocol can achieve high fidelity with a more favorable scaling of input magic states. The Clifford stage processes states efficiently with low overhead, and the non-Clifford stage completes the process with minimal additional resources, avoiding the need to process large batches in a single stage.
Solution Approach 2:
The protocol dynamically adjusts the processing by injecting magic states at various steps throughout the circuit rather than all at the beginning. This dynamic injection allows for more efficient resource utilization and reduces the peak number of input states required while maintaining high distillation fidelity.
3Ease of operation
If magic states are injected all at the start of the circuit, then the protocol is simpler, but T-gate depth and computational requirements increase
Solution Approach 1:
The protocol dynamically injects magic states at various steps throughout the circuit execution rather than all at the start. This dynamic approach distributes the computational load more evenly, reducing the T-gate depth while maintaining protocol simplicity through clear阶段性 processing in the two-stage framework.
Solution Approach 2:
The Clifford stage performs preliminary processing to prepare intermediate magic states before the non-Clifford stage completes the distillation. This preliminary action reduces the burden on subsequent stages, allowing for more efficient resource usage and reduced T-gate depth without significantly complicating the overall protocol.
Data Source
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AI summary
Disclosed herein are example embodiments of protocols to distill magic states for T-gates. Particular examples have low space overhead and use an asymptotically optimal number of input magic states to achieve a given target error. In some embodiments, the input magic states are injected at various steps in the circuit rather than all at the start of the circuit. Embodiments of the protocol can be modified to distill magic states for other gates at the third level of the Clifford hierarchy. Certain embodiments of the protocol rely on the construction of weakly self-dual Calderbank-Shor-Steane codes ("CSS codes") with many logical qubits and large distance, allowing one to implement control-Swaps on multiple qubits. Alternatively, one can use weakly-self dual CSS codes which implement controlled Hadamards for the inner code, reducing circuit depth. Several specifc small examples of this protocol are disclosed herein.