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

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