Standard quantum inference converts quantum data into classical outputs. We study an alternative inference setting in which the desired output is quantum. Such settings include quantum purity amplification (QPA), random purification and approximate cloning, and density matrix exponentiation. We show that such protocols can achieve exponentially lower sample complexity than incoherent, measurement-mediated protocols, establishing, to the best of our knowledge, the first nontrivial separations between dimension-independent coherent and dimension-dependent incoherent sample complexity. For QPA with principal-eigenstate targets and $d$-dimensional inputs, at fixed output number and spectral gap, coherent processing achieves error $\varepsilon$ using $O(1/\varepsilon)$ copies, whereas any incoherent protocol requires $Ω(d/\varepsilon)$ copies. Together, these separations motivate a unified picture of coherent quantum inference, in which an entanglement-breaking limit identifies the optimal incoherent counterpart.
Standard quantum inference converts quantum data into classical outputs. We study an alternative inference setting in which the desired output is quantum. Such settings include quantum purity amplification (QPA), random purification and approximate cloning, and density matrix exponentiation. We show that such protocols can achieve exponentially lower sample complexity than incoherent, measurement-mediated protocols, establishing, to the best of our knowledge, the first nontrivial separations between dimension-independent coherent and dimension-dependent incoherent sample complexity. For QPA with principal-eigenstate targets and d-dimensional inputs, at fixed output number and spectral gap, coherent processing achieves error ε using O(1/ε) copies, whereas any incoherent protocol requires Ω(d/ε) copies. Together, these separations motivate a unified picture of coherent quantum inference, in which an entanglement-breaking limit identifies the optimal incoherent counterpart.
Quantum purity amplification (QPA) is the task of coherently transforming n copies of a mixed state into high-fidelity copies of a chosen eigenstate. We develop a general theory of QPA for n input copies, m output copies, and arbitrary choices of target eigenstate, local dimension d, and input spectrum. We introduce the overhang-removal protocol, derive its all-site and one-site fidelities across output regimes, and show that it is optimal in several regimes, both asymptotically and nonasymptotically. For the asymptotic analysis, we use a path-graph parametrization to show that, when the target eigenvalue has a constant spectral gap Dk,min, achieving worst-case all-site error ε scales as O(m/(εDk,min2)). When m/n approaches a constant, the performance exhibits phase-like regimes, which we characterize explicitly. For the nonasymptotic analysis, we develop a theory of generalized Young diagrams that yields tight sample complexity bounds and provides the first tight dimension-uniform guarantee for optimal QPA. We also provide asymptotically efficient implementations of the overhang-removal protocol. Together, these results establish QPA as a rigorous example of coherent quantum information processing with dimension-uniform sample complexity, supplying the technical foundation for the coherent-incoherent separation developed in the companion work.
Quantum process tomography, the task of learning an unknown quantum channel from black-box access, is a central problem in quantum information. In this setting, protocols with quantum memory can coherently store and jointly process quantum information obtained from multiple channel uses, whereas protocols without quantum memory must measure after each use and retain only a classical transcript of the measurement outcomes. A fundamental open question is whether quantum memory provides a query-complexity advantage even when protocols without quantum memory may adapt their experiments based on all previous outcomes with unbounded classical computational power. In this work, we show that it does. We determine the optimal query complexity of quantum process tomography without quantum memory up to a constant factor to be Θ(din3dout3/ε2), where din and dout are the channel input and output dimensions, respectively, and ε is the target diamond-norm accuracy. More precisely, we prove that any incoherent protocol for this task, including adaptive protocols, requires Ω(din3dout3/ε2) queries, even when each channel use may be assisted by arbitrary fresh ancilla, and we present a non-adaptive, ancilla-free incoherent protocol achieving the matching upper bound O(din3dout3/ε2). Our results thereby generalize the optimal sample-complexity bounds for single-copy state tomography, recovered as the special case din=1. By contrast, coherent protocols with quantum memory achieve query complexity Θ(din2dout2/ε2). Hence, our results establish a rigorous learning separation between quantum process tomography with and without quantum memory.
Quantum purity amplification (QPA) provides a novel approach to counteracting the pervasive noise that degrades quantum states. We present the optimal QPA protocol for general quantum systems and global noise, resolving a two-decade open problem. Under strong depolarization, our protocol achieves an exponential reduction in sample complexity over the best-known methods. We provide an efficient implementation of the protocol based on generalized quantum phase estimation. Additionally, we introduce SWAPNET, a sparse and shallow circuit that enables QPA for near-term experiments. Simulations in both digital and analog quantum settings, along with experiments on superconducting quantum processors, confirm the protocol's robustness and practical utility. Our findings suggest that QPA could improve the performance of quantum information processing tasks, particularly in the context of Noisy Intermediate-Scale Quantum (NISQ) devices, where reducing the effect of noise with limited resources is critical.
While entanglement is known to enable exponential improvements in the sample complexity of quantum learning, it remains unclear which properties of entangled resources are responsible for such improvements. We address this question through the reduction criterion, a condition obeyed by all bound-entangled states. In n-qubit Pauli-channel learning, we show that restricting either the input states or the measurement effects to satisfy this criterion rules out an exponential advantage for incoherent adaptive protocols. An exponential lower bound persists for the one-sided coherent adaptive protocols considered here, even when the unrestricted side retains quantum correlations across channel uses. Using conditional min-entropy, we further quantify how the sample-complexity lower bounds weaken as larger violations of the reduction criterion are allowed. Finally, we show that the same obstruction appears in conjugate-state learning: restricted joint measurements cannot reproduce the logarithmic-sample advantage of unrestricted joint measurements on ρ⊗ρ∗. These results identify violation of the reduction criterion as a necessary condition for an exponential advantage in the learning tasks considered here.
In quantum learning tasks, quantum memory can offer exponential reductions in statistical complexity compared to any single-copy strategies, but this typically necessitates at least doubling the system size. We show that such exponential reductions can also be achieved by having access to the purification of the target mixed state. Specifically, for a low-rank mixed state, only a constant number of ancilla qubits is needed for estimating properties related to its purity, cooled form, principal component and quantum Fisher information with constant sample complexity, which utilizes single-copy measurements on the purification. Without access to the purification, we prove that these tasks require exponentially many copies of the target mixed state for any strategies utilizing a bounded number of ancilla qubits, even with the knowledge of the target state's rank. Our findings also lead to practical applications in areas such as quantum cryptography. With further discussions about the source and extent of the advantages brought by purification, our work uncovers a new resource with significant potential for quantum learning and other applications.
Along with the development of quantum technology, finding useful applications of quantum computers has been a central pursuit. Despite various quantum algorithms have been developed, many of them often require strong input assumptions, which is hardware demanding. In particular, recent advances on dequantization have revealed that the quantum advantage is more of a mere artifact of strong input assumptions. In this work, we propose a variant of these algorithms, leveraging both classical and quantum resources. Provided the classical knowledge (the entries) of the matrix/vector of interest, a classical procedure is used to pre-process this information. Then they are fed into a quantum circuit which is shown to be a block encoding of the matrix of interest. From this block-encoding, we show how to use it to tackle a wide range of problems, including principal component analysis, linear equation solving, Hamiltonian simulation, preparing ground state, and data fitting. We also analyze our protocol, showing that both the classical and quantum procedure can achieve logarithmic complexity in the input dimension, thus implying its potential for near term realization. We then discuss several implications and corollaries of our result. First,, our results suggest there are certain matrices/Hamiltonians where our method can provide exponential improvement compared to the existing ones with respect to the sparsity. Regarding dense linear systems, our method achieves exponential speed-up with respect to the inverse of error tolerance, compared to the best previously known quantum algorithm for dense systems. Last, and most importantly, regarding quantum data fitting, we show how the output of our quantum algorithms can be leveraged to predict unseen data. Thus, it provides an end-to-end application, which has been an open aspect of the previous quantum data fitting algorithm.