How many copies of a mixed state \(\rho \in \mathbb{C}^{d \times d}\) are needed to learn its spectrum? To date, the best known algorithms for spectrum estimation require as many copies as full state tomography, suggesting the possibility that learning a state's spectrum might be as difficult as learning the entire state. We show that this is not the case in the setting of unentangled measurements, by giving a spectrum estimation algorithm that uses \(n = O\left(d^{3} \cdot (\log \log(d) / \log(d))^{4}\right)\) copies of \(\rho\), which is asymptotically fewer than the \(n = \Omega(d^{3})\) copies necessary for full state tomography. Our algorithm is inspired by the technique of local moment matching from classical statistics, and shows how it can be applied in the quantum setting.

A central challenge in quantum physics is to understand the structural properties of many-body systems, both in equilibrium and out of equilibrium. For classical systems, we have a unified perspective which connects structural properties of systems at thermal equilibrium to the Markov chain dynamics that mix to them. We lack such a perspective for quantum systems: there is no framework to translate the quantitative convergence of the Markovian evolution into strong structural consequences.

We study the problem of finding a (pure) product state with optimal fidelity to an unknown $n$-qubit quantum state $ρ$, given copies of $ρ$. This is a basic instance of a fundamental question in quantum learning: is it possible to efficiently learn a simple approximation to an arbitrary state? We give an algorithm which finds a product state with fidelity $\varepsilon$-close to optimal, using $N = n^{\text{poly}(1/\varepsilon)}$ copies of $ρ$ and $\text{poly}(N)$ classical overhead. We further show that estimating the optimal fidelity is NP-hard for error $\varepsilon = 1/\text{poly}(n)$, showing that the error dependence cannot be significantly improved. For our algorithm, we build a carefully-defined cover over candidate product states, qubit by qubit, and then demonstrate that extending the cover can be reduced to approximate constrained polynomial optimization. For our proof of hardness, we give a formal reduction from polynomial optimization to finding the closest product state. Together, these results demonstrate a fundamental connection between these two seemingly unrelated questions. Building on our general approach, we also develop more efficient algorithms in three simpler settings: when the optimal fidelity exceeds $5/6$; when we restrict ourselves to a discrete class of product states; and when we are allowed to output a matrix product state.

We show that thermal states of local Hamiltonians are separable above a constant temperature. Specifically, for a local Hamiltonian$H$on a graph with degree$\mathfrak{g}$, its Gibbs state at inverse temperature$\beta$, denoted by$\rho=e^{-\beta H}/\text{tr}(e^{-\beta H})$, is a classical distribution over product states for all$\beta < 1/ (c \mathfrak{{g}})$, where$c$is a constant. This sudden death of thermal entanglement upends conventional wisdom about the presence of short-range quantum correlations in Gibbs states. Moreover, we show that we can efficiently sample from the distribution over product states. In particular, for any$\beta < 1/(c\mathfrak{g}^{3})$, we can prepare a state$\varepsilon$-close to$\rho$in trace distance with a depth-one quantum circuit and$\text{poly}(n)\log(1/\varepsilon)$classical overhead.11In independent and concurrent work, Rouzé, França, and Alhambra [37] obtain an efficient quantum algorithm for preparing high-temperature Gibbs states via a dissipative evolution.

We study the problem of Hamiltonian structure learning from real-time evolution: given the ability to apply$e^{-\mathrm{i}Ht}$for an unknown local Hamiltonian$H=\Sigma_{a=1}^{m}\lambda_{a}E_{a}$on$n$qubits, the goal is to recover$H$. This problem is already well-understood under the assumption that the interaction terms,$E_{a}$, are given, and only the interaction strengths,$\lambda_{a}$, are unknown. But how efficiently can we learn a local Hamiltonian without prior knowledge of its interaction structure? We present a new, general approach to Hamiltonian learning that not only solves the challenging structure learning variant, but also resolves other open questions in the area, all while achieving the gold standard of Heisenberg-limited scaling. In particular, our algorithm recovers the Hamiltonian to$\varepsilon$error with total evolution time$\mathcal{O}(\log(n)/\varepsilon)$, and has the following appealing properties: 1)It does not need to know the Hamiltonian terms; 2)It works beyond the short-range setting, extending to any Hamiltonian$H$where the sum of terms interacting with a qubit has bounded norm; 3)It evolves according to$H$in constant time$t$increments, thus achieving constant time resolution. As an application, we can also learn Hamiltonians exhibiting power-law decay up to accuracy$\varepsilon$with total evolution time beating the standard limit of$1/\varepsilon^{2}$.

The field of quantum machine learning (QML) produces many proposals for attaining quantum speedups for tasks in machine learning and data analysis. Such speedups can only manifest if classical algorithms for these tasks perform significantly slower than quantum ones. We study quantum-classical gaps in QML through the quantum singular value transformation (QSVT) framework. QSVT, introduced by Gilyén, Su, Low and Wiebe [GSLW19], unifies all major types of quantum speedup [MRTC21]; in particular, a wide variety of QML proposals are applications of QSVT on low-rank classical data. We challenge these proposals by providing a classical algorithm that matches the performance of QSVT in this regime up to a small polynomial overhead.

We study the problem of learning a local quantum Hamiltonian H given copies of its Gibbs state ρ = e−β H/(e−β H) at a known inverse temperature β>0. Anshu, Arunachalam, Kuwahara, and Soleimanifar gave an algorithm to learn a Hamiltonian on n qubits to precision with only polynomially many copies of the Gibbs state, but which takes exponential time. Obtaining a computationally efficient algorithm has been a major open problem, with prior work only resolving this in the limited cases of high temperature or commuting terms. We fully resolve this problem, giving a polynomial time algorithm for learning H to precision from polynomially many copies of the Gibbs state at any constant β > 0. Our main technical contribution is a new flat polynomial approximation to the exponential function, and a translation between multi-variate scalar polynomials and nested commutators. This enables us to formulate Hamiltonian learning as a polynomial system. We then show that solving a low-degree sum-of-squares relaxation of this polynomial system suffices to accurately learn the Hamiltonian.

We consider process tomography for unitary quantum channels. Given access to an unknown unitary channel acting on a d-dimensional qudit, we aim to output a classical description of a unitary that is $\varepsilon$-close to the unknown unitary in diamond norm. We design an algorithm achieving error $\varepsilon$ using $O\left(\mathrm{~d}^{2} / \varepsilon\right)$ applications of the unknown channel and only one qudit. This improves over prior results, which use $O\left(\mathrm{~d}^{3} / \varepsilon^{2}\right)$ [via standard process tomography] or $O\left(\mathrm{~d}^{2.5} / \varepsilon\right)$ [Yang, Renner, and Chiribella, PRL 2020] applications. To show this result, we introduce a simple technique to “bootstrap” an algorithm that can produce constant-error estimates to one that can produce $\varepsilon$-error estimates with the Heisenberg scaling. Finally, we prove a complementary lower bound showing that estimation requires $\Omega\left(\mathrm{d}^{2} / \varepsilon\right)$ applications, even with access to the inverse or controlled versions of the unknown unitary. This shows that our algorithm has both optimal query complexity and optimal space complexity.

Software designers in many parts of the world rely on Interactive Voice Response (IVR) and Unstructured Supplementary Service Data (USSD) to create applications that work universally on basic mobile phones. In this work, we demonstrate our use of hierarchical visualizations like icicle charts to explore the usage data generated by IVR and USSD applications, helping designers understand user behavior and evaluate the efficacy of the applications’ different components. By adding interactive elements, a viewer could zoom down to to see details about users’ behavior from a certain screen, or filter users by different demographics.

We study the problem of learning a Hamiltonian H to precision $\varepsilon$, supposing we are given copies of its Gibbs state $\rho =\exp(-\beta H)/\mathrm{Tr}(\exp(-\beta H))$ at a known inverse temperature $\beta$. Anshu, Arunachalam, Kuwahara, and Soleimanifar [AAKS21] recently studied the sample complexity (number of copies of $\rho$ needed) of this problem for geometrically local N-qubit Hamiltonians. In the high-temperature (low $\beta$) regime, their algorithm has sample complexity poly (N, 1/$\beta$, 1/$\varepsilon$) and can be implemented with polynomial, but suboptimal, time complexity. In this paper, we study the same question for a more general class of Hamiltonians. We show how to learn the coefficients of a Hamiltonian to error $\varepsilon$ with sample complexity $S=O(\log N/(\beta\varepsilon)^{2}$) and time complexity linear in the sample size, O(SN). Furthermore, we prove a matching lower bound showing that our algorithm’s sample complexity is optimal, and hence our time complexity is also optimal. In the appendix, we show that virtually the same algorithm can be used to learn H from a real-time evolution unitary $e^{-i t H}$ in a small t regime with similar sample and time complexity.

We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang’s breakthrough quantum-inspired algorithm for recommendation systems [STOC’19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén et al. [STOC’19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all prior results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: ℓ 2 -norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.

We present classical sublinear-time algorithms for solving low-rank linear systems of equations. Our algorithms are inspired by the HHL quantum algorithm for solving linear systems and the recent breakthrough by Tang of dequantizing the quantum algorithm for recommendation systems. Let $A \in \mathbb{C}^{m \times n}$ be a rank-$k$ matrix, and $b \in \mathbb{C}^m$ be a vector. We present two algorithms: a "sampling" algorithm that provides a sample from $A^{-1}b$ and a "query" algorithm that outputs an estimate of an entry of $A^{-1}b$, where $A^{-1}$ denotes the Moore-Penrose pseudo-inverse. Both of our algorithms have query and time complexity $O(\mathrm{poly}(k, κ, \|A\|_F, 1/ε)\,\mathrm{polylog}(m, n))$, where $κ$ is the condition number of $A$ and $ε$ is the precision parameter. Note that the algorithms we consider are sublinear time, so they cannot write and read the whole matrix or vectors. In this paper, we assume that $A$ and $b$ come with well-known low-overhead data structures such that entries of $A$ and $b$ can be sampled according to some natural probability distributions. Alternatively, when $A$ is positive semidefinite, our algorithms can be adapted so that the sampling assumption on $b$ is not required.

We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang’s breakthrough quantum-inspired algorithm for recommendation systems [STOC’19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén et al. [STOC’19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffices to generalize all recent results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: ℓ2-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.

We give a classical analogue to Kerenidis and Prakash’s quantum recommendation system, previously believed to be one of the strongest candidates for provably exponential speedups in quantum machine learning. Our main result is an algorithm that, given an m × n matrix in a data structure supporting certain ℓ2-norm sampling operations, outputs an ℓ2-norm sample from a rank-k approximation of that matrix in time O(poly(k)log(mn)), only polynomially slower than the quantum algorithm. As a consequence, Kerenidis and Prakash’s algorithm does not in fact give an exponential speedup over classical algorithms. Further, under strong input assumptions, the classical recommendation system resulting from our algorithm produces recommendations exponentially faster than previous classical systems, which run in time linear in m and n.