Estimating expectation values using approximate quantum states

Abstract

We introduce an approximate description of an $N$-qubit state, which contains sufficient information to estimate the expectation value of any observable to a precision that is upper bounded by the ratio of a suitably-defined seminorm of the observable to the square root of the number of the system's identical preparations $M$, with no explicit dependence on $N$. We describe an operational procedure for constructing the approximate description of the state that requires, besides the quantum state preparation, only single-qubit rotations followed by single-qubit measurements. We show that following this procedure, the cardinality of the resulting description of the state grows as $3MN$. We test the proposed method on Rigetti's quantum processor unit with 12, 16 and 25 qubits for random states and random observables, and find an excellent agreement with the theory, despite experimental errors.

8+5 pages; supersedes the general approximate state description section of arXiv:1910.10543 with improved presentation, new experiments and a discussion on related work; Version 2: minor edits; Version 3: Quantum journal version

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• Quantum Mixed State Compiling
• Statistically significant tests of multiparticle quantum correlations based on randomized measurements
• Group-theoretic error mitigation enabled by classical shadows and symmetries
• Machine learning on quantum experimental data toward solving quantum many-body problems
• Mitigating errors by quantum verification and post-selection
• Virtual distillation with noise dilution
• In the shadow of the Hadamard test: Using the garbage state for good and further modifications
• Quantifying multiparticle entanglement with randomized measurements
• Expanding variational quantum eigensolvers to larger systems by dividing the calculations between classical and quantum hardware
• Low variance estimations of many observables with tensor networks and informationally-complete measurements
• Short-time simulation of quantum dynamics by Pauli measurements
• Hardware-efficient quantum annealing with error mitigation via classical shadow
• Improving shadow estimation with locally-optimal dual frames