Estimating Fermionic Operators via Noisy Majorana Measurements
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Solution Overview
Problem
Current methods for estimating non-commuting observables in multiparticle quantum systems are hindered by high computational costs due to complex quantum circuits and large sample complexities.
Innovation Solution
A method involving a family of circuits with linearly scaling depth, differing only by the application of single qubit Pauli gates, is proposed for estimating k-body reduced fermionic density matrices and quantum Hamiltonians, utilizing simultaneous measurements of noisy Majorana fermion operators.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If complicated quantum circuits with linear depth scaling are used to estimate all k-body Fermionic reduced density matrices, then the estimation accuracy is improved, but the circuit complexity and sampling overhead increase significantly
Solution Approach 1:
The patent segments the measurement task by dividing k-body Fermionic reduced density matrices into multiple commuting subsets. Each subset can be measured simultaneously using simple quantum circuits, avoiding the need for complex linear-depth circuits while maintaining estimation accuracy. The segmentation is based on identifying commuting relationships among fermionic operators.
Solution Approach 2:
The patent applies partial action by measuring only the necessary commuting subsets of operators rather than all possible k-body operators. This reduces the total number of measurement settings required while still obtaining sufficient information for accurate estimation of the reduced density matrices.
2Device complexity
If simple quantum circuits of constant depth are used, then the circuit complexity is reduced, but the number of measurement settings increases to N^2k
Solution Approach 1:
The patent merges multiple measurement tasks by identifying and measuring commuting subsets of operators simultaneously. Instead of sequentially measuring all N^2k operators with simple circuits, the patent combines compatible operators into subsets that can be measured in parallel, reducing the total measurement time while using only constant-depth circuits.
Solution Approach 2:
The patent creates universal measurement procedures that can estimate multiple different k-body reduced density matrices using the same set of measurement settings. A single measurement configuration serves multiple estimation purposes, reducing the total number of required measurement settings from N^2k to a much smaller number.
3Reliability
If a large number of different large depth quantum circuits are performed, then the estimation of many body Hamiltonian is guaranteed, but the sampling complexity and computational cost increase
Solution Approach 1:
The patent segments the Hamiltonian estimation task into measurements of commuting operator subsets. This segmentation maintains the reliability guarantees for Hamiltonian estimation while dramatically reducing the number of circuit executions needed, as each subset can be measured efficiently and independently.
Solution Approach 2:
The patent performs preliminary analysis to identify commuting relationships among Hamiltonian operators before execution. This preliminary classification allows the design of measurement protocols that guarantee accurate estimation while minimizing the number of required circuit executions, avoiding redundant measurements.
Data Source
Figure 1a~1b
Figure 2
Figure 3a~3b
AI summary
The present invention relates to a method for estimating fermionic k-body reduced density matrices (where k=1,2) and expectation values of fermionic Hamiltonians on a quantum computer, comprising the steps of: • step i) implementing a joint measurement of noisy versions of products of Majorana fermion operators, • step ii) directly using estimates of expectation values of said products of Majorana fermion operators accessible from outputs in a post-processing of the results of the measurements performed in a computational basis on a quantum computer for estimating the fermionic k-body reduced density matrices and expectation values of fermionic Hamiltonians. The invention covers also a system comprising a quantum computer and a classical computer, programmed and configured for implementing the method. The invention covers also a computer program comprising instructions which, when the program is executed by the classical computer in the system according to invention, cause the system to carry out the method according to invention.