Summary Quantum Computing with Surface Codes arxiv.org
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The text excerpt discusses various topics related to quantum computing, including parallel modular inversion, elliptic curve point addition circuit, resource estimates for key generation, active volume architecture, and optimizations to reduce costs and improve efficiency.
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Slide Presentation (11 slides)
Key Points
- The text discusses parallel modular inversion, elliptic curve point addition circuit, resource estimates for key generation, active volume architecture, and ways to avoid the reaction limit in quantum computations.
- The importance of optimizations and balancing different factors to reduce costs and improve efficiency is highlighted.
- Algorithmic modifications, such as reusing computed states, brute force searches, and multiple modular multiplications, have resulted in significant reductions in cost and active volume.
- Active-volume architectures benefit from reduced Toffoli counts and active volumes, making them more efficient for generating keys in elliptic curve cryptography.
- Different hardware platforms have different code cycle lengths, which determine the time scale of surface-code quantum computers.
- Strict 2D-local connectivity and non-local connections can reduce the cost per key and increase computational speed in quantum architectures.
Summary
976 word summary
The text excerpt consists of figures and references from a document on quantum computing with surface codes. The figures and references are not relevant to the summary and can be omitted. The important details in the text are related to parallel modular inversion, elliptic curve point addition circuit, resource estimates for key generation, and active volume architecture. The summary can be organized into separate paragraphs for each topic.
Paragraph 1: The text excerpt includes figures showcasing parallel modular inversion of four numbers and steps in the elliptic curve point addition circuit.
Paragraph 2: The references listed in the text are not relevant to the summary.
Paragraph 3: The text mentions the motivation to explore additional instances of an algorithm to save qubits and reduce costs. It also suggests using long delay lines and additional ancilla qubits to decrease the cost of subroutines.
Paragraph 4: The text discusses the use of active volume architecture and its potential benefits in reducing the cost per key in generating elliptic curve private keys.
Paragraph 5: The text highlights the need for further optimizations in different regimes, such as balancing arithmetic and memory contributions, adjusting window size, and exploring lower-depth subroutines.
Paragraph 6: The text mentions ways to avoid the reaction limit, such as reducing reaction time, using lower-depth subroutines, and employing devices with long delay lines.
Paragraph 7: The text emphasizes the importance of reaction depth and its impact on resource estimates for quantum computations.
Paragraph 8: The text concludes by mentioning that the time per key decreases with an increase in the number of instances of an algorithm executed in parallel.
Summary: The text excerpt discusses parallel modular inversion, elliptic curve point addition circuit, resource estimates for key generation, active volume architecture, and ways to avoid the reaction limit in quantum computations. It highlights the need for optimizations and balancing different factors to reduce costs and improve efficiency. In the first scenario, the required RSG rate decreases by 27% compared to a baseline architecture. The device footprint increases by a factor of 4, but due to the cheaper inversion operation, the time per key decreases. In the second scenario, the device generates one key every 58 seconds. The device consists of 6000 qubit modules, and local connections exist for these qubits. The total error probability after executing 3000 blocks per logical cycle is below 50%. The code distance is determined in an active-volume architecture. In an active-volume architecture, the active volume per key is reduced. The cost per key is reduced by an additional 40% in the second scenario. The cost per key can be reduced by adjusting the delays. The resource estimate for the baseline architecture includes modular arithmetic operations and algorithmic modifications. The algorithmic modifications aim to reduce the resource requirements of the elliptic curve point operations. The efficient modular inversion circuit is described in detail. We have decomposed the ECC algorithm into fundamental arithmetic subroutines and made algorithmic modifications to optimize resource usage. We have introduced three modifications: reusing the state computed in the first half of the algorithm, determining 48 of the 256 bits of the key through brute force search on a classical computer, and computing multiple modular multiplications using a single inversion and a few multiplications. These modifications have resulted in significant reductions in cost and active volume. Devices with shorter code cycles have achieved a 240-fold speedup due to non-local connections and have been able to generate keys at a faster rate. The active-volume architecture has also led to a reduction in footprint. This summary provides a concise version of the text excerpt while preserving important details and highlighting key points.
Paragraph 1: Different implementations of delay lines are possible, including fiber delays and free-space delays with mirrors. The maximum usable delay line length is limited by the transmission loss rate.
Paragraph 2: A device with an active-volume architecture and a 1 ms code cycle takes 160 days to generate a 256-bit key. A more conservative approach with shorter code cycles and lower-distance surface codes could lead to a doubling of the code distance.
Paragraph 3: The resource estimates are based on approximations of Toffoli counts and active volumes. Active-volume architectures benefit from reduced Toffoli counts and active volumes, making them more efficient.
Paragraph 4: The number of logical qubits in a quantum computer depends on the total number of physical qubits and the ratio of logical qubits to physical qubits. Longer delays increase the code cycle length and the number of logical operations executed in parallel.
Paragraph 5: The physical size of a quantum computer depends on the number of resource states generated by the resource-state generators (RSGs) and the maximum number of resource states present simultaneously. The total RSG rate determines the physical size of the quantum computer.
Paragraph 6: The time scale of surface-code quantum computers is determined by the code cycle length. Different hardware platforms, such as superconducting qubits, trapped ions, and photonic fusion-based quantum computers, have different code cycle lengths.
Paragraph 7: The smaller key size of elliptic curve cryptography (ECC) makes it susceptible to potential quantum-based attacks. Resource estimates for ECC keys show that active-volume architectures can generate keys more efficiently compared to baseline architectures.
Paragraph 8: Architecture-independent gate counts and tailored optimization techniques are discussed to improve resource estimates. The cost function nQ.nTof is used to determine resource estimates for active-volume architectures.
Paragraph 9: Modifications to the architecture, such as strict 2D-local connectivity and non-local connections, can reduce the cost per key and increase computational speed. Different architectures for surface codes are compared.
Paragraph 10: Shor's algorithm is used as a case study to compute a 256-bit elliptic curve private key. An active-volume architecture with strict 2D-local connectivity can generate the key in a significantly shorter time compared to a fault-tolerant surface-code quantum computer.
Paragraph 11: The resource estimates are provided by Daniel Litinski at