Efficient quantum measurement of Pauli operators in the presence of finite sampling error

• DOI: 10.22331/q-2021-01-20-385
• Publication Date: 2021-01-20T19:20:11Z
• Authors: Ophelia Crawford; Barnaby van Straaten; Daochen Wang; Thomas Parks; Earl Campbell; Stephen Brierley
• Keywords: Pauli exclusion principle; Operator (biology); Basis (linear algebra)

Estimating the expectation value of an operator corresponding to observable is a fundamental task in quantum computation. It often impossible obtain such estimates directly, as computer restricted measuring fixed computational basis. One common solution splits into weighted sum Pauli operators and measures each separately, at cost many measurements. An improved version collects mutually commuting together before all within collection simultaneously. The effectiveness doing this depends on two factors. Firstly, we must understand improvement offered by given arrangement Paulis collections. In our work, propose natural metrics for quantifying this, operating under assumption that measurements are distributed optimally among collections so minimise overall finite sampling error. Motivated mathematical form these metrics, introduce<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mstyle mathsize="1.2em"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>S</mml:mi></mml:mrow></mml:mstyle></mml:math>ORTED<mml:math class="MJX-TeXAtom-ORD"><mml:mi>I</mml:mi></mml:mrow></mml:mstyle></mml:math>NSERTION, collecting strategy exploits weighting sum. Secondly, measure simultaneously, circuit required rotate them present efficient constructions suitably any of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="bold-italic">k</mml:mi></mml:math>independent commuting<mml:math mathvariant="bold-italic">n</mml:mi></mml:math>-qubit using most<mml:math mathvariant="bold-italic">k</mml:mi><mml:mi mathvariant="bold-italic">n</mml:mi><mml:mo mathvariant="bold">−</mml:mo><mml:mi mathvariant="bold-italic">k</mml:mi><mml:mo mathvariant="bold" stretchy="false">(</mml:mo><mml:mi mathvariant="bold">+</mml:mo><mml:mn mathvariant="bold">1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn mathvariant="bold">2</mml:mn></mml:math>and<mml:math mathvariant="bold-italic">O</mml:mi><mml:mo mathvariant="bold-italic">n</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi mathvariant="bold">log</mml:mi><mml:mo>⁡</mml:mo><mml:mi stretchy="false">)</mml:mo></mml:math>two-qubit gates respectively. Our methods numerically illustrated context Variational Quantum Eigensolver, where question molecular Hamiltonians. As measured metrics,<mml:math class="MJX-TeXAtom-ORD"><mml:mi>I</mml:mi></mml:mrow></mml:mstyle></mml:math>NSERTION outperforms four conventional greedy colouring algorithms seek minimum number