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arXiv:2601.12502 · Physical review. E

Semidefinite Programming for Quantum Channel Learning

Mikhail Gennadievich Belov, Victor Victorovich Dubov, Vadim Konstantinovich Ivanov, Alexander Yurievich Maslov, Olga Vladimirovna Proshina, Vladislav Gennadievich Malyshkin

cs.LGmath.NAquant-phSubmitted Jan 18, 2026• Announced Jan 20, 2026• 0 citations• Revised Sep 10, 2026 (v2)
v1 · Jan 18, 2026v2 · Sep 10, 2026
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Abstract

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.

Journal reference
Phys. Rev. E 114, 035302 (2026)
DOI
10.1103/kdl5-smrh

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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.
Semidefinite Programming (SDP) is a class of convex optimization programs with vast applications in control theory, quantum information, combinatorial optimization and operational research. Noisy intermediate-scale quantum (NISQ) algorithms aim to make an efficient use of the current generation of quantum hardware. However, optimizing variational quantum algorithms is a challenge as it is an NP-hard problem that in general requires an exponential time to solve and can contain many far from optimal local minima. Here, we present a current term NISQ algorithm for SDP. The classical optimization program of our NISQ solver is another SDP over a smaller dimensional ansatz space. We harness the SDP based formulation of the Hamiltonian ground state problem to design a NISQ eigensolver. Unlike variational quantum eigensolvers, the classical optimization program of our eigensolver is convex, can be solved in polynomial time with the number of ansatz parameters and every local minimum is a global minimum. Further, we demonstrate the potential of our NISQ SDP solver by finding the largest eigenvalue of up to 210002^{1000}21000 dimensional matrices and solving graph problems related to quantum contextuality. We also discuss NISQ algorithms for rank-constrained SDPs. Our work extends the application of NISQ computers onto one of the most successful algorithmic frameworks of the past few decades.
Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of εεε in poly(1/ε,input dimension)\mathrm{poly}(1/ε, \text{input dimension})poly(1/ε,input dimension) time.
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Solving optimization problems is a key task for which quantum computers could possibly provide a speedup over the best known classical algorithms. Particular classes of optimization problems including semidefinite programming (SDP) and linear programming (LP) have wide applicability in many domains of computer science, engineering, mathematics, and physics. Here we focus on semidefinite and linear programs for which the dimensions of the variables involved are exponentially large, so that standard classical SDP and LP solvers are not helpful for such large-scale problems. We propose the QSlack and CSlack methods for estimating their optimal values, respectively, which work by (1) introducing slack variables to transform inequality constraints to equality constraints, (2) transforming a constrained optimization to an unconstrained one via the penalty method, and (3) replacing the optimizations over all possible nonnegative variables by optimizations over parameterized quantum states and parameterized probability distributions. Under the assumption that the SDP and LP inputs are efficiently measurable observables, it follows that all terms in the resulting objective functions are efficiently estimable by either a quantum computer in the SDP case or a quantum or probabilistic computer in the LP case. Furthermore, by making use of SDP and LP duality theory, we prove that these methods provide a theoretical guarantee that, if one could find global optima of the objective functions, then the resulting values sandwich the true optimal values from both above and below. Finally, we showcase the QSlack and CSlack methods on a variety of example optimization problems and discuss details of our implementation, as well as the resulting performance. We find that our implementations of both the primal and dual for these problems approach the ground truth, typically achieving errors of the order 10-2.
Sometimes a mathematical tool emerges as uniquely useful and even a defining feature within some branch of physics such as Feynman diagrams. In quantum information theory, the semidefinite program (SDP) has emerged as such a tool. SDP is an optimization task in which a linear objective function is maximized over a set of Hermitian matrices with positive eigenvalues. This review discusses the highly efficient algorithms available for the SDP, and shows how comprehensively the SDP has been deployed in problems of entanglement characterization, quantum nonlocality, quantum channel capacities, and the bounding of ground-state energies.
Semidefinite programs (SDPs) are a particular class of convex optimization problems with applications in combinatorial optimization, operational research, and quantum information science. Seminal work by Brandão and Svore shows that a ``quantization'' of the matrix multiplicative-weight algorithm can provide approximate solutions to SDPs quadratically faster than the best classical algorithms by using a quantum computer as a Gibbs-state sampler. We propose a modification of this quantum algorithm and show that a similar speedup can be obtained by replacing the Gibbs-state sampler with the preparation of thermal pure quantum (TPQ) states. While our methodology incurs an additional problem-dependent error, which decreases as the problem size grows, it avoids the preparation of purified Gibbs states, potentially saving a number of ancilla qubits. In addition, we identify a spectral condition which, when met, reduces the resources further, and shifts the computational bottleneck from Gibbs state preparation to ground-state energy estimation. With classical state-vector simulations, we verify the efficiency of the algorithm for particular cases of Hamiltonian learning problems. We are able to obtain approximate solutions for two-dimensional spinless Hubbard and one-dimensional Heisenberg XXZ models for sizes of up to N=210N=2^{10}N=210 variables. For the Hubbard model, we provide an estimate of the resource requirements of our algorithm, including the number of Toffoli gates and the number of qubits.