Estimating expectation values using approximate quantum states

Marco Paini1, Amir Kalev2, Dan Padilha1, and Brendan Ruck3

1Rigetti Computing, 138 Holborn, London, EC1N 2SW, UK.
2Information Sciences Institute, University of Southern California, Arlington, VA 22203, USA.
3Rigetti Computing, 2919 Seventh St, Berkeley, CA 94710, USA.

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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.

Density matrices represent our knowledge about state of quantum systems and gives us a way to calculate expectation values and predict experimental results. Estimating the density matrix of an N-body system requires the knowledge of exponentially many numbers in terms of N. The exponentially-expensive representation of quantum states turns into practical difficulties in estimating and storing density matrices, and, moreover, raises theoretical questions about the physical meaning of quantum states. This leads to the important and natural question of whether a protocol that significantly reduces or eliminates the exponential dependence is admissible. In this work, we give an affirmative answer to the question.

Specializing to N-qubit systems as an important case study, we propose an observation-based approximate description of quantum states. The approximate description grows linearly with N (rather than exponentially in N, as for density matrices) and allow us to estimate expectation values of any observable quantity, without the need to go through the density matrix formulation, with an estimation error only dependent on the number of observations and on the observable itself, independently of N. Building the approximate description of the N-qubit state only requires repetitions of single-qubit rotations followed by single-qubit measurements. The N-independence property and the simplicity of building the approximate description has the potential for improving a wide range of quantum protocols and can be considered for implementation on today's noisy quantum computers.

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