Quantum Circuits for NISQ Matrix Trace Estimation
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Solution Overview
Problem
Classical computers face challenges in efficiently estimating the trace of large Hermitian matrices when their entries and eigenvalues are inaccessible, and existing quantum computing methods require fault-tolerant quantum computers.
Innovation Solution
A hybrid quantum-classical system using noisy intermediate-scale quantum (NISQ) processors and short-depth quantum circuits to estimate the trace of Hermitian matrices by generating random state vectors and determining moments of the matrices, enabling trace estimation without simulating the entire matrix.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If classical computers are used to estimate the trace of large Hermitian matrices, then the computational complexity increases exponentially with matrix size, but the implementation is straightforward with standard algorithms
Solution Approach 1:
The patent replaces classical computational mechanics with quantum mechanical principles. Quantum computers use qubits that can exist in superposition states, allowing simultaneous representation of multiple computational states. This substitution enables quantum algorithms to estimate matrix traces by preparing quantum states that encode matrix information and measuring observable quantities, achieving exponential speedup over classical methods for large matrices
Solution Approach 2:
The patent changes the fundamental parameter of information representation from classical bits to quantum bits (qubits). This parameter change allows the system to exploit quantum superposition and entanglement, transforming the computational approach from sequential classical operations to parallel quantum operations, thereby resolving the exponential complexity issue in trace estimation
2Productivity
If fault-tolerant quantum computers are used for trace estimation, then exponential speedup can be achieved, but the hardware requirements and system complexity increase significantly
Solution Approach 1:
The patent applies partial quantum action by using noisy intermediate-scale quantum (NISQ) processors that do not require full fault tolerance. The approach performs trace estimation with limited quantum resources and short circuit depths, accepting some noise and errors rather than requiring perfect fault tolerance. This partial application of quantum computing achieves practical speedup without the excessive hardware complexity of fault-tolerant systems
Solution Approach 2:
The patent uses NISQ processors that are currently available but have limited coherence times and higher error rates compared to future fault-tolerant machines. These quantum processors are effectively 'short-living' in terms of their operational window before decoherence, and the algorithm is designed to complete trace estimation within this limited time, avoiding the need for expensive, long-term fault-tolerant infrastructure
3Measurement precision
If full matrix simulation is performed on quantum computers, then accurate trace estimation can be achieved, but the quantum resources and computational time increase
Solution Approach 1:
The patent extracts only the necessary information for trace estimation from the full matrix by preparing specific quantum states that encode trace-related properties. Instead of simulating the entire matrix, the algorithm prepares quantum states |ψ⟩ such that measurements of certain observables directly yield trace information. This extraction approach achieves accurate trace estimation while using minimal quantum resources and computational time
Solution Approach 2:
The patent performs preliminary quantum state preparation that encodes matrix trace information in a compact form before measurement. By preparing quantum states in advance that are specifically designed to reveal trace properties upon measurement, the algorithm avoids the need for full matrix simulation during the actual estimation process, thereby reducing quantum computational time while maintaining accuracy
Data Source
AI summary
Systems and methods for operating a quantum system are described. A controller of a quantum system can generate a command signal. The quantum system can include quantum hardware having a plurality of qubits. An interface of the quantum system can control the quantum hardware based on the command signal to generate a random state vector represented by the plurality of qubits. The random state vector can include a specific number of independent entries. The interface can control the quantum hardware to determine moments of a matrix based on the random state vector. The controller can be further configured to output the moments of the matrix to a computing device to estimate a trace of the matrix using the moments.