Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2604.18884

Understanding Quantum Instruments

Akel Hashim

quant-phSubmitted Apr 20, 2026• Announced Apr 22, 2026• 0 citations• Revised Apr 24, 2026 (v2)
v1 · Apr 20, 2026v2 · Apr 24, 2026v3 · Apr 30, 2026
arXivHTMLPDF

Abstract

The quantum instrument (QI) formalism is required to model mid-circuit measurements (MCMs) and the dependence of the post-measurement state on the measurement outcome. Correctly modeling QIs is essential for applications using MCMs, such as adaptive circuits and quantum error correction. Although QIs yield a joint quantum-classical state after measurement, errors in QIs can still be represented by a $d^2 \times d^2$ superoperator (e.g., process or transfer matrix) for each outcome, just as superoperators describe Markovian errors on unitary gates. However, because the joint quantum-classical system has a distinct error model for each outcome, this complicates the usual interpretation of process- or transfer-matrix error models. This Note offers practical guidance on understanding and interpreting QI error models.

Related papers

Sequential realization of Quantum Instruments

Soham Sau, Michal Sedlák

In adaptive quantum circuits classical results of mid-circuit measurements determine the upcoming gates. This allows POVMs, quantum channels or more generally quantum instruments to be implemented sequentially, so that fewer qubits need to be used at each of the $N$ measurement steps. In this paper, we mathematically describe these problems via adaptive sequence of instruments (ASI) and show how any instrument can be decomposed into it. Number of steps $N$ and number of ancillary qubits $n_A$ needed for actual implementation are crucial parameters of any such ASI. We show an achievable lower bound on the product $N.n_A$ and we determine in which situations this tradeoff is likely to be optimal. Contrary to common intuition we show that for quantum instruments which transform $n$ to $m(>n)$ qubits, there exist $N$-step ASI implementing them just with $(m-n)$ ancillary qubits, which are remeasured $(N-1)$ times and finally used as output qubits.

quant-pharXiv:2512.14588•Dec 17, 2025•0 citations

Resource theory of interactive quantum instruments

Chung-Yun Hsieh, Armin Tavakoli, Huan-Yu Ku +1 more

Quantum instruments describe both the classical outcome and the updated quantum state in a measurement process. To do this in a non-trivial way, instruments must have the capability to interact coherently with the state that they measure. Here, we develop a resource theory for instruments. We consider a relevant quantifier of the separation between interactive and non-interactive instruments and show that it admits three distinct operational interpretations in terms of quantum information tasks. These concern (i) the preservation of maximally entangled states after a local measurement, (ii) the average ability to preserve random states after measurement, and (iii) the ability to recover the classical information generated from measuring half of a maximally entangled state. We also introduce a natural set of allowed operations and show that the third task fully characterises the resource content of instruments. Our general framework reproduces as special cases established resource theories for channels and measurements.

quant-pharXiv:2603.27676•Physical Review A•Mar 31, 2026•1 citation

Detailed, interpretable characterization of mid-circuit measurement on a transmon qubit

Piper C. Wysocki, Luke D. Burkhart, Madeline H. Morocco +12 more

Mid-circuit measurements (MCMs) are critical components of the quantum error correction protocols expected to enable utility-scale quantum computing. MCMs can be modeled by quantum instruments (a type of quantum operation or process), which can be characterized self-consistently using gate set tomography. However, experimentally estimated quantum instruments are often hard to interpret or relate to device physics. We address this challenge by adapting the error generator formalism -- previously used to interpret noisy quantum gates by decomposing their error processes into physically meaningful sums of "elementary errors" -- to MCMs. We deploy our new analysis on a transmon qubit device to tease out and quantify error mechanisms including amplitude damping, readout error, and imperfect collapse. We examine in detail how the magnitudes of these errors vary with the readout pulse amplitude, recover the key features of dispersive readout predicted by theory, and show that these features can be modeled parsimoniously using a reduced model with just a few parameters.

quant-pharXiv:2602.03938•Feb 5, 2026•0 citations

Nondisturbing Quantum Measurement Models

Stanley Gudder

A measurement model is a framework that describes a quantum measurement process. In this article we restrict attention to $MM$s on finite-dimensional Hilbert spaces. Suppose we want to measure an observable $A$ whose outcomes $A_x$ are represented by positive operators (effects) on a Hilbert Space $H$. We call $H$ the base or object system. We interact $H$ with a probe system on another Hilbert space $K$ by means of a quantum channel. The probe system contains a probe (or meter or pointer) observable $F$ whose outcomes $F_x$ are measured by an apparatus that is frequently (but need not be) classical in practice. The $MM$ protocol gives a method for determining the probability of an outcome $A_x$ for any state of $H$ in terms of the outcome $F_x$. The interaction channel usually entangles this state with an initial probe state of $K$ that can be quite complicated. However, if the channel is nondisturbing in a sense that we describe, then the entanglement is considerably simplified. In this article, we give formulas for observables and instruments measured by nondisturbing $MM$s. We begin with a general discussion of nondisturbing operators relative to a quantum context. We present two examples that illustrate this theory in terms of unitary nondisturbing channels.

arXiv:2009.12655•Sep 26, 2020•0 citations

Mitigating measurement errors in multiqubit experiments

Sergey B. Bravyi, Sarah Sheldon, Abhinav Kandala +2 more

Reducing measurement errors in multiqubit quantum devices is critical for performing any quantum algorithm. Here we show how to mitigate measurement errors by a classical postprocessing of the measured outcomes. Our techniques apply to any experiment where measurement outcomes are used for computing expected values of observables. Two error-mitigation schemes are presented based on tensor product and correlated Markovian noise models. Error rates parametrizing these noise models can be extracted from the measurement calibration data using a simple formula. Error mitigation is achieved by applying the inverse noise matrix to a probability vector that represents the outcomes of a noisy measurement. The error-mitigation overhead, including the number of measurements and the cost of the classical postprocessing, is exponential in $\ensuremath{\epsilon}n$, where $\ensuremath{\epsilon}$ is the maximum error rate and $n$ is the number of qubits. We report experimental demonstration of our error-mitigation methods on IBM Quantum devices using stabilizer measurements for graph states with $n\ensuremath{\le}12$ qubits and entangled 20-qubit states generated by low-depth random Clifford circuits.

arXiv:2006.14044•Physical Review A•Apr 9, 2021•335 citations

Measurement models and instruments

Teiko Heinosaari, Mário Ziman

Until now we have treated measurement apparatuses as devices taking quantum systems as their inputs and producing measurement outcomes as outputs. At this level of description, measurement apparatuses are fully described by observables, as discussed in Chapter 3. The measurement outcome distribution is determined by the particular input state and the observable describing the measurement device. In this chapter we explore a mathematical framework suitable for the description of measurement apparatuses for which, in addition to the measurement outcomes, the measured systems are available after the measurement is performed. In such cases we may try to obtain more information on the system by measuring some other observable, or, perhaps by filtering out some preferred output states we may use our measurement apparatus as a preparator. Clearly, in this kind of situation we need a description of measurement apparatuses that is more detailed than that of observables. Therefore, two more refined concepts, measurement models and instruments , will be investigated in this chapter. There is no customary name for a measurement model. As used here, it was first formalized in a general way by Ozawa [106], [107] who referred to the measurement process . A comprehensive general reference on quantum measurements is the monograph by Busch et al. [35], where measurement models are called premeasurements . Some other authors use the term indirect measurements . Three levels of description of measurements It is convenient to think of three levels of description when considering measurements.

Cambridge University Press eBooks•Dec 15, 2011•0 citations
The quantum instrument (QI) formalism is required to model mid-circuit measurements (MCMs) and the dependence of the post-measurement state on the measurement outcome. Correctly modeling QIs is essential for applications using MCMs, such as adaptive circuits and quantum error correction. Although QIs yield a joint quantum-classical state after measurement, errors in QIs can still be represented by a d2×d2d^2 \times d^2d2×d2 superoperator (e.g., process or transfer matrix) for each outcome, just as superoperators describe Markovian errors on unitary gates. However, because the joint quantum-classical system has a distinct error model for each outcome, this complicates the usual interpretation of process- or transfer-matrix error models. This Note offers practical guidance on understanding and interpreting QI error models.
In adaptive quantum circuits classical results of mid-circuit measurements determine the upcoming gates. This allows POVMs, quantum channels or more generally quantum instruments to be implemented sequentially, so that fewer qubits need to be used at each of the NNN measurement steps. In this paper, we mathematically describe these problems via adaptive sequence of instruments (ASI) and show how any instrument can be decomposed into it. Number of steps NNN and number of ancillary qubits nAn_AnA​ needed for actual implementation are crucial parameters of any such ASI. We show an achievable lower bound on the product N.nAN.n_AN.nA​ and we determine in which situations this tradeoff is likely to be optimal. Contrary to common intuition we show that for quantum instruments which transform nnn to m(>n)m(>n)m(>n) qubits, there exist NNN-step ASI implementing them just with (m−n)(m-n)(m−n) ancillary qubits, which are remeasured (N−1)(N-1)(N−1) times and finally used as output qubits.
Quantum instruments describe both the classical outcome and the updated quantum state in a measurement process. To do this in a non-trivial way, instruments must have the capability to interact coherently with the state that they measure. Here, we develop a resource theory for instruments. We consider a relevant quantifier of the separation between interactive and non-interactive instruments and show that it admits three distinct operational interpretations in terms of quantum information tasks. These concern (i) the preservation of maximally entangled states after a local measurement, (ii) the average ability to preserve random states after measurement, and (iii) the ability to recover the classical information generated from measuring half of a maximally entangled state. We also introduce a natural set of allowed operations and show that the third task fully characterises the resource content of instruments. Our general framework reproduces as special cases established resource theories for channels and measurements.
Mid-circuit measurements (MCMs) are critical components of the quantum error correction protocols expected to enable utility-scale quantum computing. MCMs can be modeled by quantum instruments (a type of quantum operation or process), which can be characterized self-consistently using gate set tomography. However, experimentally estimated quantum instruments are often hard to interpret or relate to device physics. We address this challenge by adapting the error generator formalism -- previously used to interpret noisy quantum gates by decomposing their error processes into physically meaningful sums of "elementary errors" -- to MCMs. We deploy our new analysis on a transmon qubit device to tease out and quantify error mechanisms including amplitude damping, readout error, and imperfect collapse. We examine in detail how the magnitudes of these errors vary with the readout pulse amplitude, recover the key features of dispersive readout predicted by theory, and show that these features can be modeled parsimoniously using a reduced model with just a few parameters.
A measurement model is a framework that describes a quantum measurement process. In this article we restrict attention to MMMMMMs on finite-dimensional Hilbert spaces. Suppose we want to measure an observable AAA whose outcomes AxA_xAx​ are represented by positive operators (effects) on a Hilbert Space HHH. We call HHH the base or object system. We interact HHH with a probe system on another Hilbert space KKK by means of a quantum channel. The probe system contains a probe (or meter or pointer) observable FFF whose outcomes FxF_xFx​ are measured by an apparatus that is frequently (but need not be) classical in practice. The MMMMMM protocol gives a method for determining the probability of an outcome AxA_xAx​ for any state of HHH in terms of the outcome FxF_xFx​. The interaction channel usually entangles this state with an initial probe state of KKK that can be quite complicated. However, if the channel is nondisturbing in a sense that we describe, then the entanglement is considerably simplified. In this article, we give formulas for observables and instruments measured by nondisturbing MMMMMMs. We begin with a general discussion of nondisturbing operators relative to a quantum context. We present two examples that illustrate this theory in terms of unitary nondisturbing channels.
Reducing measurement errors in multiqubit quantum devices is critical for performing any quantum algorithm. Here we show how to mitigate measurement errors by a classical postprocessing of the measured outcomes. Our techniques apply to any experiment where measurement outcomes are used for computing expected values of observables. Two error-mitigation schemes are presented based on tensor product and correlated Markovian noise models. Error rates parametrizing these noise models can be extracted from the measurement calibration data using a simple formula. Error mitigation is achieved by applying the inverse noise matrix to a probability vector that represents the outcomes of a noisy measurement. The error-mitigation overhead, including the number of measurements and the cost of the classical postprocessing, is exponential in $\ensuremath{\epsilon}n$, where $\ensuremath{\epsilon}$ is the maximum error rate and nnn is the number of qubits. We report experimental demonstration of our error-mitigation methods on IBM Quantum devices using stabilizer measurements for graph states with $n\ensuremath{\le}12$ qubits and entangled 20-qubit states generated by low-depth random Clifford circuits.
Until now we have treated measurement apparatuses as devices taking quantum systems as their inputs and producing measurement outcomes as outputs. At this level of description, measurement apparatuses are fully described by observables, as discussed in Chapter 3. The measurement outcome distribution is determined by the particular input state and the observable describing the measurement device. In this chapter we explore a mathematical framework suitable for the description of measurement apparatuses for which, in addition to the measurement outcomes, the measured systems are available after the measurement is performed. In such cases we may try to obtain more information on the system by measuring some other observable, or, perhaps by filtering out some preferred output states we may use our measurement apparatus as a preparator. Clearly, in this kind of situation we need a description of measurement apparatuses that is more detailed than that of observables. Therefore, two more refined concepts, measurement models and instruments , will be investigated in this chapter. There is no customary name for a measurement model. As used here, it was first formalized in a general way by Ozawa [106], [107] who referred to the measurement process . A comprehensive general reference on quantum measurements is the monograph by Busch et al. [35], where measurement models are called premeasurements . Some other authors use the term indirect measurements . Three levels of description of measurements It is convenient to think of three levels of description when considering measurements.