Magic State Distillation with Low Space Overhead and Staged Injection
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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 arbitrary fidelity, necessitating a large number of input magic states, which is inefficient and costly in terms of resources.
Innovation Solution
The proposed protocol uses weakly self-dual Calderbank-Shor-Steane codes to implement control-Swaps and measure properties of magic states, allowing for asymptotically optimal input count and reduced space overhead by injecting magic states at various steps in the circuit, rather than all at once, and adapts to distill magic states for other gates in the Clifford hierarchy.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional magic state distillation protocols are used, then arbitrary fidelity can be achieved, but space overhead and input magic state count become excessively large
Solution Approach 1:
The protocol segments the distillation process into multiple stages with different code distances, where each stage processes a subset of input magic states. This segmentation allows the system to achieve high fidelity output without requiring all input states to be processed simultaneously, thereby reducing the peak space overhead and input count requirements compared to conventional single-stage protocols.
Solution Approach 2:
The protocol introduces a temporal dimension to the distillation process by injecting magic states at various steps rather than all at once. This transforms the problem from a spatial resource constraint (all states needed simultaneously) to a temporal processing sequence, allowing asymptotically optimal input count while maintaining manageable space overhead through the use of CSS codes with specific dimensional properties.
2Reliability
If concatenation is used to achieve arbitrary fidelity, then error suppression improves, but device complexity and space overhead increase
Solution Approach 1:
The protocol employs dynamic code selection where the code distance and structure are adapted to the specific distillation stage and error rate requirements. Rather than fixed concatenation layers, the system dynamically chooses appropriate CSS codes for each stage, optimizing the balance between error suppression and device complexity. This dynamic approach allows high-order error suppression without the rigid complexity of traditional concatenation schemes.
Solution Approach 2:
The protocol changes key parameters such as code distance, block size, and injection timing across different distillation stages. By varying these parameters rather than using fixed concatenation structures, the system achieves asymptotically optimal performance with reduced space overhead. The parameter changes allow the protocol to adapt to different fidelity requirements without proportionally increasing device complexity.
3Ease of operation
If all magic states are injected at the start of the circuit, then the protocol is simple to implement, but space overhead increases
Solution Approach 1:
The protocol prepares and injects magic states at specific preliminary steps before they are needed for the actual distillation operations. This preliminary action allows the system to stage the availability of magic states, reducing the peak space overhead by ensuring states are present only when required rather than loading all states at the beginning. The preliminary injection timing is carefully coordinated with the distillation circuit stages.
Solution Approach 2:
The protocol employs periodic injection of magic states at regular intervals throughout the circuit execution rather than a single bulk injection. This periodic action distributes the space requirements over time, allowing the system to maintain lower peak memory usage while still providing sufficient input states for the distillation process. The periodic injection rhythm is synchronized with the circuit depth and error suppression requirements.
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.