Ewin Tang

Ewin Tang, John Wright, Mark Zhandry

We give a natural problem over input quantum oracles $U$ which cannot be solved with exponentially many black-box queries to $U$ and $U^\dagger$, but which can be solved with constant many queries to $U$ and $U^*$, or $U$ and $U^{\mathrm{T}}$. We also demonstrate a quantum commitment scheme that is secure against adversaries that query only $U$ and $U^\dagger$, but is insecure if the adversary can query $U^*$. These results show that conjugate and transpose queries do give more power to quantum algorithms, lending credence to the idea put forth by Zhandry that cryptographic primitives should prove security against these forms of queries. Our key lemma is that any circuit using $q$ forward and inverse queries to a state preparation unitary for a state $σ$ can be simulated to $\varepsilon$ error with $n = \mathcal{O}(q^2/\varepsilon)$ copies of $σ$. Consequently, for decision tasks, algorithms using (forward and inverse) state preparation queries only ever perform quadratically better than sample access. We also identify a motif, which we call the "acorn trick", where generically strengthening a quantum resource can be possible if the output is allowed to be random, bypassing no-go theorems for deterministic algorithms. We demonstrate this idea for several settings, including controlization and purification.

Ainesh Bakshi, Allen Liu, Ankur Moitra et al.

We study the problem of learning a local quantum Hamiltonian $H$ given copies of its Gibbs state $ρ= e^{-βH}/\textrm{tr}(e^{-βH})$ at a known inverse temperature $β>0$. Anshu, Arunachalam, Kuwahara, and Soleimanifar (arXiv:2004.07266) gave an algorithm to learn a Hamiltonian on $n$ qubits to precision $ε$ 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 [Alhambra'22 (arXiv:2204.08349)], [Anshu, Arunachalam'22 (arXiv:2204.08349)], with prior work only resolving this in the limited cases of high temperature [Haah, Kothari, Tang'21 (arXiv:2108.04842)] or commuting terms [Anshu, Arunachalam, Kuwahara, Soleimanifar'21]. We fully resolve this problem, giving a polynomial time algorithm for learning $H$ to precision $ε$ 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.

Arjan Cornelissen, Joao F. Doriguello, Alessandro Luongo et al.

Clustering is one of the most important tools for analysis of large datasets, and perhaps the most popular clustering algorithm is Lloyd's algorithm for $k$-means. This algorithm takes $n$ vectors $V=[v_1,\dots,v_n]\in\mathbb{R}^{d\times n}$ and outputs $k$ centroids $c_1,\dots,c_k\in\mathbb{R}^d$; these partition the vectors into clusters based on which centroid is closest to a particular vector. We present a classical $\varepsilon$-$k$-means algorithm that performs an approximate version of one iteration of Lloyd's algorithm with time complexity $\tilde{O}\big(\frac{\|V\|_F^2}{n}\frac{k^{2}d}{\varepsilon^2}(k + \log{n})\big)$, exponentially improving the dependence on the data size $n$ and matching that of the "$q$-means" quantum algorithm originally proposed by Kerenidis, Landman, Luongo, and Prakash (NeurIPS'19). Moreover, we propose an improved $q$-means quantum algorithm with time complexity $\tilde{O}\big(\frac{\|V\|_F}{\sqrt{n}}\frac{k^{3/2}d}{\varepsilon}(\sqrt{k}+\sqrt{d})(\sqrt{k} + \log{n})\big)$ that quadratically improves the runtime of our classical $\varepsilon$-$k$-means algorithm in several parameters. Our quantum algorithm does not rely on quantum linear algebra primitives of prior work, but instead only uses QRAM to prepare simple states based on the current iteration's clusters and multivariate quantum amplitude estimation. Finally, we provide classical and quantum query lower bounds, showing that our algorithms are optimal in most parameters.

Ewin Tang, John Wright

Many quantum algorithms, to compute some property of a unitary $U$, require access not just to $U$, but to $cU$, the unitary with a control qubit. We show that having access to $cU$ does not help for a large class of quantum problems. For a quantum circuit which uses $cU$ and $cU^\dagger$ and outputs $|ψ(U)\rangle$, we show how to "decontrol" the circuit into one which uses only $U$ and $U^\dagger$ and outputs $|ψ(φU)\rangle$ for a uniformly random phase $φ$, with a small amount of time and space overhead. When we only care about the output state up to a global phase on $U$, then the decontrolled circuit suffices. Stated differently, $cU$ is only helpful because it contains global phase information about $U$. A version of our procedure is described in an appendix of Sheridan, Maslov, and Mosca (arXiv:0810.3843). Our goal with this work is to popularize this result by generalizing it and investigating its implications, in order to counter negative results in the literature which might lead one to believe that decontrolling is not possible. As an application, we give a simple proof for the existence of unitary ensembles which are pseudorandom under access to $U$, $U^\dagger$, $cU$, and $cU^\dagger$.

Ewin Tang, John Wright

We prove that the generic quantum speedups for brute-force search and counting only hold when the process we apply them to can be efficiently inverted. The algorithms speeding up these problems, amplitude amplification and amplitude estimation, assume the ability to apply a state preparation unitary $U$ and its inverse $U^\dagger$; we give problem instances based on trace estimation where no algorithm which uses only $U$ beats the naive, quadratically slower approach. Our proof of this is simple and goes through the compressed oracle method introduced by Zhandry. Since these two subroutines are responsible for the ubiquity of the quadratic "Grover" speedup in quantum algorithms, our result explains why such speedups are far harder to come by in the settings of quantum learning, metrology, and sensing. In these settings, $U$ models the evolution of an experimental system, so implementing $U^\dagger$ can be much harder -- tantamount to reversing time within the system. Our result suggests a dichotomy: without inverse access, quantum speedups are scarce; with it, quantum speedups abound.

Ainesh Bakshi, Allen Liu, Ankur Moitra et al.

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 $H = \sum_{a = 1}^m λ_a E_a$ 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, $λ_a$, 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 $O(\log (n)/\varepsilon)$, 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 $1/\varepsilon^2$.

Jeongwan Haah, Robin Kothari, Ewin Tang

We study the problem of learning a Hamiltonian $H$ to precision $\varepsilon$, supposing we are given copies of its Gibbs state $ρ=\exp(-βH)/\operatorname{Tr}(\exp(-βH))$ at a known inverse temperature $β$. Anshu, Arunachalam, Kuwahara, and Soleimanifar (Nature Physics, 2021, arXiv:2004.07266) recently studied the sample complexity (number of copies of $ρ$ needed) of this problem for geometrically local $N$-qubit Hamiltonians. In the high-temperature (low $β$) regime, their algorithm has sample complexity poly$(N, 1/β,1/\varepsilon)$ 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 $S = O(\log N/(β\varepsilon)^{2})$ and time complexity linear in the sample size, $O(S N)$. 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 $e^{-it H}$ in a small $t$ regime with similar sample and time complexity.

Nai-Hui Chia, András Gilyén, Tongyang Li et al.

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, Su, Low, and Wiebe [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 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: $\ell^2$-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.

Ewin Tang

A central roadblock to analyzing quantum algorithms on quantum states is the lack of a comparable input model for classical algorithms. Inspired by recent work of the author [E. Tang, STOC'19], we introduce such a model, where we assume we can efficiently perform $\ell^2$-norm samples of input data, a natural analogue to quantum algorithms that assume efficient state preparation of classical data. Though this model produces less practical algorithms than the (stronger) standard model of classical computation, it captures versions of many of the features and nuances of quantum linear algebra algorithms. With this model, we describe classical analogues to Lloyd, Mohseni, and Rebentrost's quantum algorithms for principal component analysis [Nat. Phys. 10, 631 (2014)] and nearest-centroid clustering [arXiv:1307.0411]. Since they are only polynomially slower, these algorithms suggest that the exponential speedups of their quantum counterparts are simply an artifact of state preparation assumptions.

Ewin Tang

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 $m \times n$ matrix in a data structure supporting certain $\ell^2$-norm sampling operations, outputs an $\ell^2$-norm sample from a rank-$k$ approximation of that matrix in time $O(\text{poly}(k)\log(mn))$, 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 $n$. The main insight of this work is the use of simple routines to manipulate $\ell^2$-norm sampling distributions, which play the role of quantum superpositions in the classical setting. This correspondence indicates a potentially fruitful framework for formally comparing quantum machine learning algorithms to classical machine learning algorithms.