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 $\mathcal{U}$ (partial unitarity). The constructed operator $\mathcal{U}$ 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.
The problem of an optimal mapping between Hilbert spaces IN of ∣\ensuremathψ\ensuremath⟩ and OUT of ∣\ensuremathϕ\ensuremath⟩ based on a set of wavefunction measurements (within a phase) \ensuremathψl\ensuremath→\ensuremathϕl,l=1,\ensuremath⋯,M, is formulated as an optimization problem maximizing the total fidelity \ensuremath∑l=1M\ensuremathω(l)∣\ensuremath⟨\ensuremathϕl∣U∣\ensuremathψl\ensuremath⟩∣2 subject to probability preservation constraints on U (partial unitarity). The constructed operator U can be considered as an IN to OUT quantum channel; it is a partially unitary rectangular matrix (an isometry) of dimension dim(OUT)\ifmmode×\else\texttimes\fidim(IN) transforming operators as AOUT=UAINU\ifmmode†\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.
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 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 Bs. 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 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 Bs 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.
The Uhlmann fidelity F(ρ0,ρ1)=tr∣ρ0ρ1∣ is one of the most fundamental quantities in quantum information theory for quantifying the closeness between two quantum states. Estimating the Uhlmann fidelity to within additive error ε requires a number of copies of the states, or queries to their state-preparation circuits, that depends at least linearly on the smaller of the ranks of ρ0 and ρ1. Consequently, this rank dependence disappears when either state is pure, in which case the query and sample complexities depend only polynomially on 1/ε. However, the known optimal estimator for F(ρ,∣ψ⟩⟨ψ∣) due to Fang and Wang (ESA 2025) requires prior knowledge of which state is pure.
In this work, we remove this mathematically unnecessary prior-knowledge requirement and establish an optimal estimator for F(ρ,∣ψ⟩⟨ψ∣) under the sole promise that one of the two states is pure, without knowing which one. Our estimator is obtained by specializing the refined algorithmic Uhlmann transform of Utsumi, Nakata, Wang, and Takagi (2025) to the case where one state is pure. In this setting, the Uhlmann fidelity can be recovered as follows: apply a unitary dilation of trA(∣ψ0⟩⟨ψ1∣) (or its inverse) to the reference register R of the purification ∣ψ1⟩ (or ∣ψ0⟩) on the registers A and R, estimate the corresponding square-root amplitude in each case, and take the maximum of the resulting two estimates.
Roughly speaking, a symmetry of some mathematical object is an invertible transformation that leaves all relevant structure as it is. Thus a symmetry of a set is just a bijection (as sets have no further structure, whence invertibility is the only demand on a symmetry), a symmetry of a topological space is a homeomorphism, a symmetry of a Banach space is a linear isometric isomorphism, and, crucially important for this chapter, a symmetry of a Hilbert space H is a unitary operator , i.e., a linear map u:H→H satisfying one and hence all of the following equivalent conditions: uu∗=u∗u=1H ; u is invertible with u−1=u∗ ; u is a surjective isometry (or, if dim(H)<∞ , just an isometry); u is invertible and preserves the inner product, i.e., ⟨uφ,uψ⟩=⟨φ,ψ⟩(φ,ψ∈H). These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Earlier, we proved a duality between two optimizations problems [Phys. Rev. Lett. 104, 120501 (2010)]. The primary one is, given two quantum channels M and N, to find a quantum channel R such that $\mathcal{R}\ensuremath{\circ}\mathcal{N}$ is optimally close to 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 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, 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. 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 study the separation of geometric effects from quantum interference in quantum optimization algorithms. Constrained optimization problems such as routing, assignment, and scheduling are often encoded as product spaces of local variables, together with global feasibility penalties. The central algorithmic question we address is how a constraint-preserving mixing operator transports quantum amplitude across an exponential search space in the presence of local and global constraints. We develop a framework that separates three effects that are usually intermixed: amplitude transport, coherent interference among transported amplitudes, and problem-dependent classical postprocessing. We show that the mixing operator alone does not have a target-seeking ability. Concretely, the normalized distribution induced by its amplitude transport moves toward the distance profile of a uniformly random configuration. Thus, quantum sampling advantage may only arise when the phases of the many computational paths reaching a target configuration are sufficiently aligned for their amplitudes to reinforce. We show that, when the cost phases are engineered so that these paths add coherently, a number of circuit alternations growing only logarithmically with problem size suffices to convert the sum of their absolute contributions into a lower bound on the target amplitude, yielding a certified success probability independent of the ambient Hilbert-space dimension, the search-space size, or the feasible-set cardinality. We develop applications to problem-specific transpilation diagnostics, scalable hardware probes, constraint-induced classical maps of quantum-generated samples, the attribution of solution quality between the quantum distribution and classical post-processing in hybrid quantum-classical workflows and connections to distance-partitioned product spaces from classical coding theory.