Quantum State Purity Measurement Beyond Full State Tomography
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Solution Overview
Problem
Existing quantum computing technologies face challenges in measuring quantum state purity efficiently, particularly for systems with a large number of qubits, as full state tomography scales exponentially, making it intractable beyond a small number of qubits.
Innovation Solution
A method for determining quantum state purity using raw measurement data from quantum hardware benchmarking experiments, employing statistical analysis of probability distributions to calculate variance and apply the Porter-Thomas distribution, allowing for accurate purity measurement with a fixed number of experiments.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If full state tomography is used to measure quantum state purity, then measurement precision is improved, but device complexity and resource requirements scale exponentially with the number of qubits
Solution Approach 1:
The patent segments the quantum state measurement problem into individual probability measurements of computational basis states. Instead of measuring the complete density matrix through full tomography, the method divides the task into obtaining probabilities P(i) for each basis state |i⟩, then calculates purity from these segmented measurements using the formula Tr(ρ²) = Σ P(i)². This segmentation reduces measurement complexity from exponential to polynomial scaling.
Solution Approach 2:
The patent extracts only the essential information needed for purity measurement from the complete quantum state. By taking out just the diagonal elements (probabilities of basis states) rather than the full density matrix, the method obtains sufficient data to calculate purity without the exponential overhead of complete state tomography. The extracted probabilities are sufficient because purity depends only on the distribution of occupation probabilities across basis states.
2Productivity
If the number of qubits is increased to enhance computing power, then productivity is improved, but the feasibility of full state tomography deteriorates due to exponential scaling
Solution Approach 1:
The patent changes the measurement parameter from complete density matrix elements to only the occupation probabilities of computational basis states. This parameter change allows scalability because measuring probabilities of basis states requires only polynomial resources in the number of qubits, whereas full tomography requires exponential resources. The purity calculation Tr(ρ²) = Σ P(i)² uses only these changed parameters, enabling the method to scale to larger quantum systems.
3Measurement precision
If additional experiments are conducted to improve measurement accuracy, then measurement precision is improved, but loss of time increases
Solution Approach 1:
The patent applies partial action by measuring only what is necessary for purity determination. Instead of performing complete state tomography which requires measurements of all density matrix elements, the method performs partial measurements of only the diagonal elements (basis state probabilities). This partial measurement approach provides sufficient information for purity calculation while reducing the number of required experiments and associated time loss.
Data Source
AI summary
Methods, systems and apparatus for measuring quantum state purity. In one aspect, a method for determining an average purity of multiple output quantum states, wherein the multiple output quantum states correspond to applications of respective random quantum circuits of a same circuit depth to a same initial quantum state, the method including: obtaining a plurality of data items, wherein each data item corresponds to a respective random quantum circuit of the same circuit depth and represents a probability that application of the respective random quantum circuit to the initial quantum state produces a respective measurement result; calculating a variance of a plurality of data items; determining a Porter-Thomas distribution having a dimension equal to a dimension of each output quantum state; and dividing the calculated variance by a variance of the Porter-Thomas distribution to determine the average purity.