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arXiv:2407.04406 · Physical review. E

Quantum channel learning

Mikhail Gennadievich Belov, Victor V. Dubov, Alexey Vladimirovich Filimonov, Vladislav Gennadievich Malyshkin

Published Jan 3, 2025• 8 citations
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Abstract

The problem of an optimal mapping between Hilbert spaces IN and OUT, based on a series of density matrix mapping measurements ${\ensuremath{\rho}}^{(l)}\ensuremath{\rightarrow}{\ensuremath{\varrho}}^{(l)}$, $l=1\ensuremath{\cdots}M$, is formulated as an optimization problem maximizing the total fidelity $\mathcal{F}={\ensuremath{\sum}}_{l=1}^{M}{\ensuremath{\omega}}^{(l)}F({\ensuremath{\varrho}}^{(l)},{\ensuremath{\sum}}_{s}{B}_{s}{\ensuremath{\rho}}^{(l)}{B}_{s}^{\ifmmode\dagger\else\textdagger\fi{}})$ subject to probability preservation constraints on Kraus operators ${B}_{s}$. For $F(\ensuremath{\varrho},\ensuremath{\sigma})$ in the form that total fidelity can be represented as a quadratic form with superoperator $\mathcal{F}={\ensuremath{\sum}}_{s}\ensuremath{\langle}{B}_{s}|S|{B}_{s}\ensuremath{\rangle}$ (either exactly or as an approximation) an iterative algorithm is developed. The work introduces two important generalizations of unitary learning: (1) IN/OUT states are represented as density matrices; (2) the mapping itself is formulated as a mixed unitary quantum channel ${A}^{OUT}={\ensuremath{\sum}}_{s}{|{w}_{s}|}^{2}{\mathcal{U}}_{s}{A}^{IN}{\mathcal{U}}_{s}^{\ifmmode\dagger\else\textdagger\fi{}}$ (no general quantum channel yet). This marks a crucial advancement from the commonly studied unitary mapping of pure states ${\ensuremath{\phi}}_{l}=\mathcal{U}{\ensuremath{\psi}}_{l}$ to a quantum channel, which allows us to distinguish probabilistic mixture of states and their superposition. An application of the approach is demonstrated on unitary learning of density matrix mapping ${\ensuremath{\varrho}}^{(l)}=\mathcal{U}{\ensuremath{\rho}}^{(l)}{\mathcal{U}}^{\ifmmode\dagger\else\textdagger\fi{}}$, in this case a quadratic on $\mathcal{U}$ fidelity can be constructed by considering $\sqrt{{\ensuremath{\rho}}^{(l)}}\ensuremath{\rightarrow}\sqrt{{\ensuremath{\varrho}}^{(l)}}$ mapping, and on a quantum channel, where quadratic on ${B}_{s}$ fidelity is an approximation---a quantum channel is then obtained as a hierarchy of unitary mappings, a mixed unitary channel. The approach can be applied to studying quantum inverse problems, variational quantum algorithms, quantum tomography, and more. A software product implementing the algorithm is available from the authors.

DOI
10.1103/physreve.111.015302

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Earlier, we proved a duality between two optimizations problems [Phys. Rev. Lett. 104, 120501 (2010)]. The primary one is, given two quantum channels M\mathcal{M}M and N\mathcal{N}N, to find a quantum channel R\mathcal{R}R such that $\mathcal{R}\ensuremath{\circ}\mathcal{N}$ is optimally close to M\mathcal{M}M as measured by the worst-case entanglement fidelity. The dual problem involves the information obtained by the environment through the so-called complementary channels $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{M}}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{N}}$, and consists in finding a quantum channel ${\mathcal{R}}^{\ensuremath{'}}$ such that ${\mathcal{R}}^{\ensuremath{'}}\ensuremath{\circ}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{M}}$ is optimally close to $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{N}}$. It turns out to be easier to find an approximate solution to the dual problem in certain important situations, notably when M\mathcal{M}M is the identity channel---the problem of quantum error correction---yielding a good near-optimal worst-case entanglement fidelity as well as the corresponding near-optimal correcting channel. Here we provide more detailed proofs of these results. In addition, we generalize the main theorem to the case where there are certain constraints on the implementation of R\mathcal{R}R, namely, on the number of Kraus operators. We also offer a simple algebraic form for the near-optimal correction channel in the case M=id\mathcal{M}=\mathrm{id}M=id. For approximate error correction, we show that any $\ensuremath{\varepsilon}$-correctable channel is, up to appending an ancilla, $\ensuremath{\varepsilon}$-close to an exactly correctable one. We also demonstrate an application of our theorem to the problem of minimax state discrimination.
We address the problem of optimally approximating the action of a desired and unavailable quantum channel $\mathrm{\ensuremath{\Phi}}$ having at our disposal a single use of a given set of other channels ${{\mathrm{\ensuremath{\Psi}}}_{i}}$. The problem is recast to look for the least distinguishable channel from $\mathrm{\ensuremath{\Phi}}$ among the convex set ${\ensuremath{\sum}}_{i}{p}_{i}{\mathrm{\ensuremath{\Psi}}}_{i}$, and the corresponding optimal weights pi{{p}_{i}}pi​ provide the optimal convex mixing of the available channels ${{\mathrm{\ensuremath{\Psi}}}_{i}}$. For single-qubit channels we study specifically cases where the available convex set corresponds to covariant channels or to Pauli channels, and the desired target map is an arbitrary unitary transformation or a generalized damping channel.
The problem of reconstructing a quantum channel from a sample of classical data is considered. When the total fidelity can be represented as a ratio of two quadratic forms (e.g., in the case of mapping a mixed state to a pure state, projective operators, unitary learning, and others), Semidefinite Programming (SDP) can be applied to solve the fidelity optimization problem with respect to the Choi matrix. A remarkable feature of SDP is that the optimization is convex, which allows the problem to be efficiently solved by a variety of numerical algorithms. We have tested several commercially available SDP solvers, all of which allowed for the reconstruction of quantum channels of different forms. A notable feature is that the Kraus rank of the obtained quantum channel typically comprises less than a few percent of its maximal possible value. This suggests that a relatively small Kraus rank quantum channel is typically sufficient to describe experimentally observed classical data. The theory was also applied to the problem of reconstructing projective operators from data. Finally, we discuss a classical computational model based on quantum channel transformation, performed and calculated on a classical computer, possibly hardware-optimized.
The subject of this thesis, quantum estimation theory, is inspired by recent developments, both practical and theoretical, in the new field of quantum information and computation. The main task of quantum estimation theory, the subject of this thesis, is that given many identical copies of an unknown quantum system one wants to guess or estimate what the system is. This is done by performing an measurement and analyzing the data obtained from it. A quantum state is represented by a density matrix. In some cases one does not need to know the whole density matrix, sometimes a partial characterization is enough. Chapter 2 of this thesis is about optimally estimating the spectrum, i.e. the eigenvalues, of a density matrix. This is useful, for example, for estimating the entanglement of a bipartite pure state. The results of chapter 2 tell us that one can optimally estimate bipartite entanglement with LOCC measurements on one of the two parties. Unitary operations transform density matrices to density matrices. They are the gates of quantum information processing. Suppose one has one such unitary operation but one does not know what it does. In order to characterize it one needs to know how it transforms any input state. In order to know how a particular state is transformed, one can prepare that state, feed it to the unitary many times, and measure the output. One state is, of course, not enough. It turns out that the unitary can be completely characterized with a finite set of states, this is known as quantum process tomography. Unfortunately, in many cases it is difficult to have available in the lab all the input states needed to perform quantum process tomography. It is here where entanglement comes to the rescue. One pair of maximally entangled qubits, where one of them is used as input for the unitary gate and the other is left untouched, is sufficient. All the information about the unknown gate is contained in the joint output state. This is the problem treated in chapter 3. Now the output state needs to be measured. Again, one could measure the two qubits separately (LOCC) or jointly (collective measurement). Is there anything to gain by performing joint measurements? The answer is yes, optimal collective measurements are at least 3 times more accurate than any LOCC measurement. Chapter 4 treats the problem of estimating commuting unitary gates. There it is shown that this extra information makes it unnecessary to use an entangled input state and to use collective measurements at the output. In chapters 3 and 4 it is assumed that only one copy of the unitary gate is available. If, however, more than one copy (say N ) are available at the same time one could use an N-fold entangled state as input. If no entanglement is used, classical statistics tells us that the error of the estimation is proportional to 1/√N. In the literature, it has been shown that by using an entangled state in the input one can achieve an error proportional to 1/N. However, this has been shown only for qubit gates. In chapter 5 important steps are taken in the direction of showing the same thing for a general unitary gate, but this remains an open problem.
The problem of an optimal mapping between Hilbert spaces IN of $|\ensuremath{\psi}\ensuremath{\rangle}$ and OUT of $|\ensuremath{\phi}\ensuremath{\rangle}$ based on a set of wavefunction measurements (within a phase) ${\ensuremath{\psi}}_{l}\ensuremath{\rightarrow}{\ensuremath{\phi}}_{l}, l=1,\ensuremath{\cdots},M$, is formulated as an optimization problem maximizing the total fidelity ${\ensuremath{\sum}}_{l=1}^{M}{\ensuremath{\omega}}^{(l)}|\ensuremath{\langle}{\ensuremath{\phi}}_{l}|\mathcal{U}|{\ensuremath{\psi}}_{l}{\ensuremath{\rangle}|}^{2}$ subject to probability preservation constraints on U\mathcal{U}U (partial unitarity). The constructed operator U\mathcal{U}U can be considered as an IN to OUT quantum channel; it is a partially unitary rectangular matrix (an isometry) of dimension $dim(\mathrm{OUT})\ifmmode\times\else\texttimes\fi{}dim(\mathrm{IN})$ transforming operators as ${A}^{\mathrm{OUT}}=\mathcal{U}{A}^{\mathrm{IN}}{\mathcal{U}}^{\ifmmode\dagger\else\textdagger\fi{}}$. An iterative algorithm for finding the global maximum of this optimization problem is developed, and its application to a number of problems is demonstrated. A software product implementing the algorithm is available from the authors.