We study quantum process tomography given the prior information that the map is a unitary or close to a unitary process. We show that a unitary map on a $d$ -level system is completely characterized by a minimal set of ${d}^{2}+d$ elements associated with a collection of POVMs, in contrast to the ${d}^{4}\ensuremath{-}{d}^{2}$ elements required for a general completely positive trace-preserving map. To achieve this lower bound, one must probe the map with particular sets of $d$ pure states. We further compare the performance of different estimators inspired by compressed sensing, to reconstruct a near-unitary process from such data. We find that when we have accurate prior information, an appropriate compressed sensing method reduces the required data needed for high-fidelity estimation, and different estimators applied to the same data are sensitive to different types of noise. Convex optimization inspired by the techniques of compressed sensing can therefore be used both as indicators of error models and to validate the use of the prior assumptions.We study quantum process tomography given the prior information that the map is a unitary or close to a unitary process. We show that a unitary map on a $d$-level system is completely characterized by a minimal set of ${d}^{2}+d$ elements associated with a collection of POVMs, in contrast to the ${d}^{4}\ensuremath{-}{d}^{2}$ elements required for a general completely positive trace-preserving map. To achieve this lower bound, one must probe the map with particular sets of $d$ pure states. We further compare the performance of different estimators inspired by compressed sensing, to reconstruct a near-unitary process from such data. We find that when we have accurate prior information, an appropriate compressed sensing method reduces the required data needed for high-fidelity estimation, and different estimators applied to the same data are sensitive to different types of noise. Convex optimization inspired by the techniques of compressed sensing can therefore be used both as indicators of error models and to validate the use of the prior assumptions.

arXiv:1404.2877Physical Review AJul 17, 201487 citations

Near-Optimal Parallel Unitary Process Tomography in Diamond Distance

Noam Scully, Sisi Zhou

Unitary process tomography is the act of learning how an unknown unitary gate acts on a quantum system. It is an essential procedure for realizing a functioning quantum computer. As quantum technologies develop, it is important to understand the computational resources needed to conduct unitary process tomography using a variety of types of circuits. In particular, when measuring error in diamond distance, it is known how to achieve optimal query complexity using a sequential circuit architecture, but it had not been known whether a parallel circuit architecture could achieve the same optimal query complexity. In this work, we prove that unitary process tomography with near-optimal query complexity is indeed possible using a parallel circuit by introducing an explicit, novel parallel procedure. This closes the gap between sequential and parallel circuits up to a logarithmic factor, showing that parallel and sequential strategies are nearly equally capable for this task.Unitary process tomography is the act of learning how an unknown unitary gate acts on a quantum system. It is an essential procedure for realizing a functioning quantum computer. As quantum technologies develop, it is important to understand the computational resources needed to conduct unitary process tomography using a variety of types of circuits. In particular, when measuring error in diamond distance, it is known how to achieve optimal query complexity using a sequential circuit architecture, but it had not been known whether a parallel circuit architecture could achieve the same optimal query complexity. In this work, we prove that unitary process tomography with near-optimal query complexity is indeed possible using a parallel circuit by introducing an explicit, novel parallel procedure. This closes the gap between sequential and parallel circuits up to a logarithmic factor, showing that parallel and sequential strategies are nearly equally capable for this task.

quant-pharXiv:2609.36127Sep 30, 20260 citations

Tomography of Quantum States in Small Dimensions

Markus Grassl

Electronic Notes in Discrete MathematicsJun 19, 200578 citations

High-dimensional quantum gates using full-field spatial modes of photons

Florian Brandt, Markus Hiekkamäki, Frédéric Bouchard +2 more

Unitary transformations are the fundamental building blocks of gates and operations in quantum information processing, allowing the complete manipulation of quantum systems in a coherent manner. In the case of photons, optical elements that can perform unitary transformations are readily available only for some degrees of freedom, e.g., wave plates for polarization. However, for high-dimensional states encoded in the transverse spatial modes of light, performing arbitrary unitary transformations remains a challenging task for both theoretical proposals and actual implementations. Following the idea of multi-plane light conversion, we show that it is possible to perform a broad variety of unitary operations at high quality by using only a few phase modulation planes. More importantly, we experimentally implement several high-dimensional quantum gates for up to five-dimensional states encoded in the full-field mode structure of photons. In particular, we realize cyclic and quantum Fourier transformations, known as Pauli X ^ -gates and Hadamard H ^ -gates, respectively, with an average visibility of more than 90%. In addition, we demonstrate near-perfect “unitarity” by means of quantum process tomography, unveiling a process purity of 99%. Last, we demonstrate the benefit of the two independent spatial degrees of freedom, i.e., azimuthal and radial, and implement a two-qubit controlled-NOT quantum operation on a single photon. Thus, our demonstrations open up new paths to implement high-dimensional quantum operations, which can be applied to various tasks in quantum communication, computation, and sensing schemes.Unitary transformations are the fundamental building blocks of gates and operations in quantum information processing, allowing the complete manipulation of quantum systems in a coherent manner. In the case of photons, optical elements that can perform unitary transformations are readily available only for some degrees of freedom, e.g., wave plates for polarization. However, for high-dimensional states encoded in the transverse spatial modes of light, performing arbitrary unitary transformations remains a challenging task for both theoretical proposals and actual implementations. Following the idea of multi-plane light conversion, we show that it is possible to perform a broad variety of unitary operations at high quality by using only a few phase modulation planes. More importantly, we experimentally implement several high-dimensional quantum gates for up to five-dimensional states encoded in the full-field mode structure of photons. In particular, we realize cyclic and quantum Fourier transformations, known as Pauli X ^ -gates and Hadamard H ^ -gates, respectively, with an average visibility of more than 90%. In addition, we demonstrate near-perfect “unitarity” by means of quantum process tomography, unveiling a process purity of 99%. Last, we demonstrate the benefit of the two independent spatial degrees of freedom, i.e., azimuthal and radial, and implement a two-qubit controlled-NOT quantum operation on a single photon. Thus, our demonstrations open up new paths to implement high-dimensional quantum operations, which can be applied to various tasks in quantum communication, computation, and sensing schemes.

arXiv:1907.13002OpticaDec 23, 2019177 citations

Tomography of quantum detectors

Jeff S. Lundeen, Alejandro Feito, Hendrik B. Coldenstrodt-Ronge +6 more

arXiv:0807.2444Nature PhysicsNov 16, 2008364 citations

We study quantum process tomography given the prior information that the map is a unitary or close to a unitary process. We show that a unitary map on a

d

-level system is completely characterized by a minimal set of

{d}^{2}+d

elements associated with a collection of POVMs, in contrast to the ${d}^{4}\ensuremath{-}{d}^{2}$ elements required for a general completely positive trace-preserving map. To achieve this lower bound, one must probe the map with particular sets of

d

pure states. We further compare the performance of different estimators inspired by compressed sensing, to reconstruct a near-unitary process from such data. We find that when we have accurate prior information, an appropriate compressed sensing method reduces the required data needed for high-fidelity estimation, and different estimators applied to the same data are sensitive to different types of noise. Convex optimization inspired by the techniques of compressed sensing can therefore be used both as indicators of error models and to validate the use of the prior assumptions.