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.
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.
Many central tasks in fundamental physics and quantum information processing are possible only insofar as mixed quantum states can be made purer. In this work, we prove that the linearity and positivity of quantum mechanics impose general restrictions on quantum purification, unveiling a new fundamental principle of quantum information processing. We first establish that no quantum operation can transform a finite number of copies of an unknown quantum state or channel into an exactly pure output that depends non-trivially on the input, thereby ruling out an important form of universal purification in both static and dynamical settings. Building on this, we show that, upon relaxing the requirement of exact purity, one can establish quantitative sample-complexity lower bounds for approximate purification that hold for arbitrary physically allowed strategies, whose scaling matches the performance of purification-related tasks across several different areas of quantum information processing. Moreover, this lower bound leads to a generalized standard quantum limit for learning arbitrary functions of a quantum state, greatly extending earlier results based on quantum Fisher information and revealing a deep connection between purification and quantum learning. Extending this principle to other important settings, we establish, for the first time, an exponential sample-complexity lower bound for approximate pure dilation state preparation and a no-go theorem for approximate bosonic Gaussian state purification with passive Gaussian operations, establishing much more stringent limitations under practical operational constraints.
Quantum resources enable us to achieve an exponential advantage in learning the properties of unknown physical systems by employing quantum memory. While entanglement with quantum memory is recognized as a necessary qualitative resource, its quantitative role remains less understood. In this work, we distinguish between two fundamental resources provided by quantum memory—entanglement and ancilla qubits—and analyze their separate contributions to the sampling complexity of quantum learning. Focusing on the task of Pauli channel learning, a prototypical example of quantum channel learning, remarkably, we prove that vanishingly small entanglement in the input state already suffices to accomplish the learning task with only a polynomial number of channel queries in the number of qubits. In contrast, we show that without a sufficient number of ancilla qubits, even learning partial information about the channel demands an exponentially large sample complexity. Thus, our findings reveal that while a large amount of entanglement is not necessary, the dimension of the quantum memory is a crucial resource. Hence, by identifying how the two resources contribute differently, our work offers deeper insight into the nature of the quantum learning advantage. While it is known that entanglement plays a role in learning the properties of unknown physical systems, the detailed quantitative dependence is unclear. Here, the authors investigate the fundamental resources behind quantum advantage in channel learning, revealing that while extensive entanglement is unnecessary, the number of ancilla qubits is crucial.
Learning properties of quantum states from measurement data is a fundamental challenge in quantum information. The sample complexity of such tasks depends crucially on the measurement primitive. While shadow tomography achieves sample-efficient learning by allowing entangling measurements across many copies, it requires prohibitively deep circuits. At the other extreme, two-copy measurements already yield exponential advantages over single-copy strategies in tasks such as Pauli tomography. In this work we show that such sharp separations extend far beyond the two-copy regime: for every prime c we construct explicit learning tasks of degree c, which are exponentially hard with (c - 1)-copy measurements but efficiently solvable with c-copy measurements. Our protocols are not only sample-efficient but also realizable with shallow circuits. Extending further, we show that such finite-degree tasks exist for all square-free integers c, pointing toward a general principle underlying their existence. Together, our results reveal an infinite hierarchy of multi-copy learning problems, uncovering new phase transitions in sample complexity and underscoring the role of reliable quantum memory as a key resource for exponential quantum advantage.
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.
We study the power of quantum memory for learning properties of quantum systems and dynamics, which is of great importance in physics and chemistry. Many state-of-the-art learning algorithms require access to an additional external quantum memory. While such a quantum memory is not required a priori, in many cases, algorithms that do not utilize quantum memory require much more data than those which do. We show that this trade-off is inherent in a wide range of learning problems. Our results include the following: •We show that to perform shadow tomography on ann-qubit stateρwithMobservables, any algorithm without quantum memory requiresΩ~(min(M,2n))samples ofρin the worst case. Up to log factors, this matches the upper bound of [1], and completely resolves an open question in [2], [3]. •We establish exponential separations between algorithms with and without quantum memory for purity testing, distinguishing scrambling and depolarizing evolutions, and uncovering symmetry in physical dynamics. Our separations improve and generalize prior work of [4] by allowing for a broader class of algorithms without quantum memory. •We give the first tradeoff between quantum memory and sample complexity. More precisely, we prove that to estimate absolute values of alln-qubit Pauli observables, algorithms withk<nqubits of quantum memory require at leastΩ(2(n−k)/3)samples, but there is an algorithm usingn-qubit quantum memory which only requiresO(n)samples. The separations we show are sufficiently large and could already be evident, for instance, with tens of qubits. This provides a concrete path towards demonstrating real-world advantage for learning algorithms with quantum memory.
We study stabilizer state testing and learning with limited coherent quantum memory. Here an algorithm sequentially receives copies of an unknown n-qubit state, but may keep only k qubits of coherent quantum memory between measurements. With unrestricted memory, seminal work of Gross, Nezami and Walter showed how to test n-qubit stabilizer states using 6 copies, which is dimension independent, unlike the learning complexity of Θ(n). We show that this testing-vs-learning separation is lost under memory constraints. More concretely we show that
(1) The sample complexity of testing stabilizer states in the k-qubit memory framework is Θ(n−k). Our upper bound goes via a novel connection to the hidden shift problem and the lower bound is proven using a novel approach to average case bounds on likelihood ratios via combinatorics of the stochastic orthogonal group.
(2) The sample complexity of learning stabilizer states with k qubits of memory, in the non-adaptive framework, is Θ(n2/k).
As a further application of our techniques, we prove an exponential lower bound for purity testing even when the memory may be left coherent throughout the protocol. Our main results identify coherent quantum memory as the resource enabling the usual separation between stabilizer testing and learning. In particular, even with k=0.99n qubits of memory, there is no constant-copy stabilizer tester; furthermore for k=cn qubits of memory (for 0<c<1), stabilizer testing is as hard as learning, with both requiring Θ(n) copies.