The ability of quantum computers to directly manipulate and analyze quantum states stored in quantum memory allows them to learn about aspects of our physical world that would otherwise be invisible given a modest number of measurements. Here we investigate a new learning resource, which could be available to quantum computers in the future—measurements on the unknown state accompanied by its complex conjugate ρ ⊗ ρ ∗ . For a certain shadow tomography task, we surprisingly find that measurements on only copies of ρ ⊗ ρ ∗ can be exponentially more powerful than measurements on ρ ⊗ K , even for large K . This expands the class of provable exponential advantages using only a constant overhead quantum memory, or minimal quantum memory, and we provide a number of examples where the state ρ ∗ is naturally available in both computational and physical applications. In addition, we precisely quantify the power of classical shadows on single copies under a generalized Clifford ensemble and give a class of quantities that can be efficiently learned. The learning task we study in both the single copy and quantum memory settings is physically natural and corresponds to real-space observables with a limit of bosonic modes, where it achieves an exponential improvement in detecting certain signals under a noisy background. We quantify a new and powerful resource in quantum learning, and we believe the advantage may find applications in improving quantum simulation, learning from quantum sensors, and uncovering new physical phenomena. Published by the American Physical Society 2024
The ability of quantum computers to directly manipulate and analyze quantum states stored in quantum memory allows them to learn about aspects of our physical world that would otherwise be invisible given a modest number of measurements. Here we investigate a new learning resource, which could be available to quantum computers in the future—measurements on the unknown state accompanied by its complex conjugate ρ ⊗ ρ ∗ . For a certain shadow tomography task, we surprisingly find that measurements on only copies of ρ ⊗ ρ ∗ can be exponentially more powerful than measurements on ρ ⊗ K , even for large K . This expands the class of provable exponential advantages using only a constant overhead quantum memory, or minimal quantum memory, and we provide a number of examples where the state ρ ∗ is naturally available in both computational and physical applications. In addition, we precisely quantify the power of classical shadows on single copies under a generalized Clifford ensemble and give a class of quantities that can be efficiently learned. The learning task we study in both the single copy and quantum memory settings is physically natural and corresponds to real-space observables with a limit of bosonic modes, where it achieves an exponential improvement in detecting certain signals under a noisy background. We quantify a new and powerful resource in quantum learning, and we believe the advantage may find applications in improving quantum simulation, learning from quantum sensors, and uncovering new physical phenomena. Published by the American Physical Society 2024
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: (1) We show that to perform shadow tomography on an n-qubit state rho with M observables, any algorithm without quantum memory requires Ω(min(M,2n)) samples of rho in the worst case. Up to logarithmic factors, this matches the upper bound of [HKP20] and completely resolves an open question in [Aar18, AR19]. (2) We establish exponential separations between algorithms with and without quantum memory for purity testing, distinguishing scrambling and depolarizing evolutions, as well as uncovering symmetry in physical dynamics. Our separations improve and generalize prior work of [ACQ21] by allowing for a broader class of algorithms without quantum memory. (3) We give the first tradeoff between quantum memory and sample complexity. We prove that to estimate absolute values of all n-qubit Pauli observables, algorithms with k<n qubits of quantum memory require at least Ω(2(n−k)/3) samples, but there is an algorithm using n-qubit quantum memory which only requires O(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.
Recent results have established dramatic advantages in learning properties of quantum states when a quantum computer is available to process or jointly measure multiple copies of the unknown quantum state. Learning tasks can be accomplished with exponentially fewer copies of the state when compared to optimized classical learning strategies that are restricted to measuring one copy of the state at a time. While these results were established in abstract settings and for artificial learning tasks, they motivate the application of quantum computers to imaging and sensing of weak electromagnetic fields since these settings are ultimately concerned with the learning of unknown quantum states. In this work we apply these new results in quantum learning to the problem of learning Gaussian states of the electromagnetic field, which are germane since they describe most fields used in imaging and sensing. In order to connect with quantum learning theory, we consider the transduction of an n-mode Gaussian state into a register of qubits on a quantum computer followed by optimized measurements on these qubits to extract the parameters defining the original Gaussian state. We rigorously bound the number of copies of the Gaussian state required to achieve worst-case additive error in parameter estimates. The scaling of this bound with n is exponentially better than naïve strategies for characterizing Gaussian states and matches recently derived bounds for characterization of Gaussian states using continuous-variable (CV) classical shadows. In addition, our bound has a polynomially better dependence on the energy of the multimode Gaussian state compared to the CV shadows protocol.
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.
Learning about physical systems from quantum-enhanced experiments can outperform learning from experiments in which only classical memory and processing are available. Whereas quantum advantages have been established for state learning, quantum process learning is less understood. We establish an exponential quantum advantage for learning an unknown n -qubit quantum process N . We show that a quantum memory allows to efficiently solve the following tasks: (a) learning the Pauli transfer matrix (PTM) of an arbitrary N , (b) predicting expectation values of Pauli-sparse observables measured on the output of an arbitrary N upon input of a Pauli-sparse state, and (c) predicting expectation values of arbitrary observables measured on the output of an unknown N with sparse PTM upon input of an arbitrary state. With quantum memory, these tasks can be solved using linearly-in- n many copies of the Choi state of N . In contrast, any learner without quantum memory requires exponentially-in- n many queries, even when using adaptively designed experiments. In proving this separation, we extend existing shadow tomography bounds from states to channels. Moreover, we combine PTM learning with polynomial interpolation to learn arbitrary Hamiltonians from short-time dynamics. Our results highlight the power of quantum-enhanced experiments for learning highly complex quantum dynamics.
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.
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.