dorsal/arxiv
View SchemaOptimal Quantum Measurements of Expectation Values of Observables
| Authors | Emanuel Knill, Gerardo Ortiz, Rolando D. Somma |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0607019 |
| URL | https://arxiv.org/abs/quant-ph/0607019 |
| DOI | 10.1103/PhysRevA.75.012328 |
Abstract
Experimental characterizations of a quantum system involve the measurement of expectation values of observables for a preparable state |psi> of the quantum system. Such expectation values can be measured by repeatedly preparing |psi> and coupling the system to an apparatus. For this method, the precision of the measured value scales as 1/sqrt(N) for N repetitions of the experiment. For the problem of estimating the parameter phi in an evolution exp(-i phi H), it is possible to achieve precision 1/N (the quantum metrology limit) provided that sufficient information about H and its spectrum is available. We consider the more general problem of estimating expectations of operators A with minimal prior knowledge of A. We give explicit algorithms that approach precision 1/N given a bound on the eigenvalues of A or on their tail distribution. These algorithms are particularly useful for simulating quantum systems on quantum computers because they enable efficient measurement of observables and correlation functions. Our algorithms are based on a method for efficiently measuring the complex overlap of |psi> and U|psi>, where U is an implementable unitary operator. We explicitly consider the issue of confidence levels in measuring observables and overlaps and show that, as expected, confidence levels can be improved exponentially with linear overhead. We further show that the algorithms given here can typically be parallelized with minimal increase in resource usage.
{
"annotation_id": "21b6cb99-8431-42f5-b4ed-46fe7756fb9d",
"date_created": "2026-03-02T18:02:27.178000Z",
"date_modified": "2026-03-02T18:02:27.178000Z",
"file_hash": "5e8eb53871d93d42a3c278b287e9ae4fa926431ca57dd8a0bb6adcdd5126154c",
"private": false,
"record": {
"abstract": "Experimental characterizations of a quantum system involve the measurement of\nexpectation values of observables for a preparable state |psi\u003e of the quantum\nsystem. Such expectation values can be measured by repeatedly preparing |psi\u003e\nand coupling the system to an apparatus. For this method, the precision of the\nmeasured value scales as 1/sqrt(N) for N repetitions of the experiment. For the\nproblem of estimating the parameter phi in an evolution exp(-i phi H), it is\npossible to achieve precision 1/N (the quantum metrology limit) provided that\nsufficient information about H and its spectrum is available. We consider the\nmore general problem of estimating expectations of operators A with minimal\nprior knowledge of A. We give explicit algorithms that approach precision 1/N\ngiven a bound on the eigenvalues of A or on their tail distribution. These\nalgorithms are particularly useful for simulating quantum systems on quantum\ncomputers because they enable efficient measurement of observables and\ncorrelation functions. Our algorithms are based on a method for efficiently\nmeasuring the complex overlap of |psi\u003e and U|psi\u003e, where U is an implementable\nunitary operator. We explicitly consider the issue of confidence levels in\nmeasuring observables and overlaps and show that, as expected, confidence\nlevels can be improved exponentially with linear overhead. We further show that\nthe algorithms given here can typically be parallelized with minimal increase\nin resource usage.",
"arxiv_id": "quant-ph/0607019",
"authors": [
"Emanuel Knill",
"Gerardo Ortiz",
"Rolando D. Somma"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.75.012328",
"title": "Optimal Quantum Measurements of Expectation Values of Observables",
"url": "https://arxiv.org/abs/quant-ph/0607019"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "bee3e77c-48ca-45e3-a475-e7cde570d15a",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}