We show that prior knowledge that a state of several photons can be represented in compact form in an unknown basis enables the recovery of the quantum state from partial measurements in a single setup.
Recovery of quantum states from measurements is an essential component in quantum information processing. In quantum optical systems, which naturally offer low decoherence and easy manipulation, quantum states are characterized by correlation measurements. When the states comprise more photons so as to encode more qubits, high-order correlation measurements are required. However, high-order correlations are hard to measure in experiments, as the rate of high-order coincidences decreases very fast when increasing the correlation order. This results in a poor signal-to-noise ratio. Likewise, the number of measurements required to characterize a quantum state increases exponentially with the increase in the number of qubits. Here, we use structure, present in most quantum states of interest (for quantum computing, cryptography, boson sampling, etc.), to recover the full quantum state of three photons from two-fold correlations in a single experimental setup.
We present an efficient protocol for learning an unknown k-local Lindblad generator on n qubits using only product-state preparations, short-time evolution, and single-qubit Pauli measurements, without prior knowledge of the interaction structure. For fixed k and a supplied weighted interaction-strength bound αˉ, the protocol estimates all Hamiltonian and dissipative Pauli--GKSL coefficients with total coefficient error at each site at most ε with probability at least 1−δ using Ok(αˉ2n2k−2ε−2log(n/δ)) samples and polylogarithmically many distinct evolution times. This guarantee requires no degree, sparsity, or tail promise. A convex optimization converts these estimates into a valid k-local Lindblad generator with diamond-norm error at most ε using Ok(αˉ2n2kε−2log(n/δ)) samples and polynomial-time classical postprocessing. With a supplied local effective-sparsity bound r and a sufficiently small sitewise subthreshold tail Ok(ε), coefficient recovery requires only Ok(αˉ2r2ε−2log(n/δ)) samples, without a supplied threshold or coefficient locations and without a coefficient-gap assumption. In particular, exactly sparse bounded-degree models have logarithmic sample dependence on n. We also provide guard-band support recovery and complementary supplied-candidate guarantees, extend the guarantees to model misspecification, and prove complementary sample-complexity lower bounds.
This paper presents a method for identifying unknown coefficients of basis elements in an effective Hamiltonian of single-qubit cases.This is done directly from the measurement statistics of a quantum system's unitary evolution, utilizing a unique method of Lie algebra coefficient recovery.The method exploits the fact that repeated application of a unitary operator amplifies the generator coefficient linearly in the exponent.The result produces measurement probabilities that constrain the unknown coefficient differently at each power tested.Inversion of any single measurement probability yields a multivalued set of candidates due to the periodic nature of the trigonometric functions involved.However, the true coefficient can be identified as the unique true value satisfying a consistency criterion across all powers simultaneously.The method is demonstrated numerically for the single-generator Hamiltonian and extended analytically to the two-generator case where geometric (𝐻 𝑒 = 𝑎𝑋) (𝐻 𝑒 = 𝑎𝑋 + 𝑏𝑍), conjugation reduces the Hamiltonian to an effective single-generator form.Numerical simulations further verify the geometric reduction and the subsequent coefficient reconstruction for representative two-generator Hamiltonians, given a known rotation angle.In both cases, the correct coefficients are recovered from measurement statistics, without ancilla qubits, coherent control circuits, or prior knowledge of the system's spectrum; for the two-generator case, the rotation angle used to align the Hamiltonian is currently taken as given rather than estimated from data.
We study the problem of learning a sparse quantum state, an n-qubit quantum state whose density matrix has at most s nonzero matrix entries in an unknown product basis. While such states admit compact classical descriptions, they can carry long-range entanglement that prevents reconstruction from local reduced density matrices alone. Therefore, previous learning approaches addressed such long-range-entangled states using many entangling gates to extract the necessary information. In this work, we show that sparse states can nevertheless be efficiently learned using only single-qubit measurements. Specifically, when the sparsity s is constant, our algorithm can learn sparse states from single-qubit measurements with polynomial sample complexity and classical computational complexity. When s grows polynomially with n, sparse states can still be learned from single-qubit measurements with polynomial sample complexity, although efficient classical computation is not guaranteed in general. In this regime, however, the classical computational complexity becomes quasipolynomial when the state is sparse in an unknown basis that is a product of a known fixed finite set of single-qubit bases (e.g., eigenbases of Pauli operators). These results establish efficient learning of sparse states with long-range entanglement without entangling gates, and the single-qubit measurement requirements make our algorithms compatible with current quantum devices.