Quantum Circuit Complexity Reduction via Overlapping Block Decomposition

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Solution Overview

Problem

Current quantum algorithms for simulating real-time evolution of local Hamiltonians on lattices have a gate count of Õ(n^2) for constant time and accuracy, which is inefficient compared to the linear scaling expected for spacetime volume, and lack a clear understanding of error analysis in Lie-Trotter-Suzuki expansions.

Innovation Solution

Decomposing the real-time evolution operator into overlapping smaller blocks of unitary operators based on Lieb-Robinson bounds, with the size of overlap proportional to the logarithm of the number of qubits, allows for a quantum circuit that achieves gate complexity matching Nature's linear scaling up to logarithmic factors, and adapts to time-dependent Hamiltonians.

Engineering Contradictions & Design Principles

VSEngineering Contradiction Analysis

1Productivity

If quantum algorithms use standard Lie-Trotter-Suzuki expansions for real-time evolution, then the evolution can be simulated, but the gate count scales as Õ(n^2) which is inefficient

Engineering Contradiction:

Improvesimulation efficiencyVSAvoidgate count

Core Design Contradiction:

ProductivityVSDevice complexity

Solution Approach 1:

The patent segments the time evolution operator into overlapping spatial blocks of size O(log n) qubits each. By dividing the lattice into such blocks and using Lieb-Robinson bounds to control the error from neglecting long-range interactions, the method achieves O(Tn polylog(Tn/ε)) gate complexity, improving upon the Õ(n^2) scaling of standard methods.

Inventive Principle:

Principle #1Segmentation

2Measurement precision

If the overlap size between blocks is increased to improve accuracy, then the simulation precision improves, but the circuit depth and gate count increase

Engineering Contradiction:

Improvesimulation accuracyVSAvoidcircuit depth

Core Design Contradiction:

Measurement precisionVSDevice complexity

Solution Approach 1:

The patent optimizes the block size parameter to be O(log n) qubits, which balances the trade-off between accuracy and circuit complexity. This parameter choice ensures that the overlap is sufficient to capture relevant correlations while keeping the number of gates and circuit depth manageable, achieving O(T polylog(Tn/ε)) depth scaling.

Inventive Principle:

Principle #35Parameter changes

3Productivity

If quantum algorithms aim for linear scaling in spacetime volume, then the efficiency matches Nature's scaling, but current algorithms lack clear error analysis in Lie-Trotter-Suzuki expansions

Engineering Contradiction:

Improvespacetime volume scalingVSAvoiderror analysis clarity

Core Design Contradiction:

ProductivityVSReliability

Solution Approach 1:

The patent uses Lieb-Robinson bounds to provide rigorous error analysis for the block decomposition method. By bounding the speed of information propagation in quantum lattice systems, the method controls the error from neglecting long-range interactions between blocks, enabling reliable linear scaling in spacetime volume with quantifiable accuracy.

Inventive Principle:

Principle #23Feedback

Data Source

PatentEP3711003B1Quantum simulation of real time evolution of lattice hamiltonians

Publication Date: 2023.08.02 MICROSOFT TECHNOLOGY LICENSING LLC

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AI summary

Embodiments of the disclosed technology concern a quantum circuit configured to implement a real time evolution unitary of a Hamiltonian in a quantum computing device, wherein a unit time evolution unitary operator is decomposed into overlapping smaller blocks of unitary operators. In some implementations, (a) the size of the overlap is proportional to the logarithm of a number of qubits in the simulated system, (b) the size of the overlap is proportional to the logarithm of a total simulated evolution time, and/or (c) the size of the overlap is proportional to a Lieb-Robinson velocity.