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A hybrid quantum classical (HQC) computer, which includes both a classical computer component and a quantum computer component, implements improvements to expectation value estimation in quantum circuits, in which the number of shots to be performed in order to compute the estimation is reduced by applying a quantum circuit that imposes an orbital rotation to the quantum state during each shot instead of applying single-qubit context-selection gates. The orbital rotations are determined through the decomposition of a Hamiltonian or another objective function into a set of orbital frames. The variationally minimized expectation value of the Hamiltonian or the other objective function may then be used to determine the extent of an attribute of the system, such as the value of a property of the electronic structure of a molecule, chemical compound, or other extended system.

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**Independent Claim 1.**A method for using a measurement module to compute an expectation value of a first operator more efficiently than Pauli-based grouping, the first operator comprising a plurality of component operators, wherein at least one of the plurality of component operators is not a product of Pauli operators, the method comprising: 1) computing the expectation value of the first operator, comprising: (a) on a quantum computer, using the measurement module to make a quantum measurement of at least one of the plurality of component operators, to produce a plurality of measurement outcomes of the at least one of the plurality of component operatorsand (b) on a classical computer, computing the expectation value of the first operator by averaging at least some of the plurality of measurement outcomes.**Independent Claim 18.**A system for using a measurement module to compute an expectation value of a first operator more efficiently than Pauli-based grouping, the first operator comprising a plurality of component operators, wherein at least one of the plurality of component operators is not a product of Pauli operators, the system comprising: a quantum computer comprising the measurement module, wherein the measurement module is adapted to make a quantum measurement of at least one of the plurality of component operators, to produce a plurality of measurement outcomes of the at least one of the plurality of component operatorsand a classical computer comprising at least one processor and at least one non-transitory computer-readable medium comprising computer program instructions which, when executed by the at least one processor, cause the at least one processor to compute the expectation value of the operator by averaging at least some of the plurality of measurement outcomes.**Independent Claim 35.**A method for computing an expectation value of a first operator more efficiently than Pauli-based grouping, the first operator comprising a plurality of component operators, wherein at least one of the plurality of component operators is not a product of Pauli operators, the method performed by a classical computer comprising at least one processor and at least one non-transitory computer-readable medium comprising computer program instructions executable by the at least one processor to perform the method, the method comprising: 1) computing the expectation value of the first operator, comprising: (a) simulating a quantum computer measurement module to make a simulated quantum measurement of at least one of the plurality of component operators, to produce a plurality of measurement outcomes of the at least one of the plurality of component operatorsand (b) computing the expectation value of the first operator by averaging at least some of the plurality of measurement outcomes.**Independent Claim 38.**A system for computing an expectation value of a first operator more efficiently than Pauli-based grouping, the first operator comprising a plurality of component operators, wherein at least one of the plurality of component operators is not a product of Pauli operators, the system comprising at least one non-transitory computer-readable medium comprising computer program instructions executable by at least one processor to perform a method, the method comprising: 1) computing the expectation value of the first operator, comprising: (a) simulating a quantum computer measurement module to make a simulated quantum measurement of at least one of the plurality of component operators, to produce a plurality of measurement outcomes of the at least one of the plurality of component operatorsand (b) computing the expectation value of the first operator by averaging at least some of the plurality of measurement outcomes.