Magic State Distillation Using CSS Codes With Low Space Overhead
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Solution Overview
Problem
Current magic state distillation protocols for quantum computing require high space overhead and involve concatenation of small routines to achieve low error fidelity, necessitating a large number of input magic states, which is inefficient and costly in terms of resources.
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, allowing for asymptotically optimal input count and low space overhead, with the ability to adapt for other gates in the Clifford hierarchy, reducing the need for concatenation and improving error suppression.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If current magic state distillation protocols are used, then error suppression is achieved, but space overhead is high and input count is large
Solution Approach 1:
The protocol segments the distillation process into modular components using CSS codes with specific parameters [[n,k,d]]. Each code instance processes a subset of input magic states, allowing systematic error suppression while controlling resource usage. The segmentation enables scalable design where larger codes provide higher error suppression without linearly increasing all resources.
Solution Approach 2:
The protocol utilizes families of CSS codes with varying parameters (n, k, d) to achieve different error suppression levels. By selecting appropriate code parameters, one can optimize the balance between input magic states required and output error rate achieved. The parameter d (distance) directly controls error suppression capability while n and k control resource overhead.
2Reliability
If concatenation of small routines is used to achieve low error fidelity, then error suppression improves, but device complexity increases
Solution Approach 1:
The protocol employs universal CSS code structures that can distill magic states for any gate in the third level of the Clifford hierarchy. This multi-functionality eliminates the need for separate specialized routines for different gates, reducing overall device complexity while maintaining high error fidelity through systematic code application.
Solution Approach 2:
The CSS codes are designed to be self-correcting through their inherent error detection and correction capabilities. The codes automatically identify and correct errors in input magic states without requiring external verification routines, simplifying the protocol structure while achieving high output fidelity.
3Reliability
If more physical qubits are used, then error suppression increases, but space overhead increases
Solution Approach 1:
The protocol applies different CSS code parameters to different portions of the distillation process based on local requirements. Codes with higher distance d are applied where maximum error suppression is needed, while codes with lower overhead are used where resource constraints are tighter. This localized optimization achieves high error suppression without uniformly increasing physical qubit count across all operations.
Data Source
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. The space overhead, defined as the ratio between the physical qubits to the number of output magic states, is asymptotically constant, while both the number of input magic states used per output state and the T-gate depth of the circuit scale linearly in the logarithm of the target error. Unlike other distillation protocols, examples of the disclosed protocol achieve this performance without concatenation and 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, with the same asymptotic performance. 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. This code is referred to herein as the “inner code”. The control-Swaps are then used to measure properties of the magic state and detect errors, using another code that is referred to as the “outer code”. Alternatively, one can use weakly-self dual CSS codes which implement controlled Hadamards for the inner code, reducing circuit depth. Several specific small examples of this protocol are disclosed herein.