arXiv subjects

Sathyawageeswar Subramanian

Publications and source records attributed to Sathyawageeswar Subramanian.

At least 19 recordsLinked to original sources

arXiv2026

Average-case hardness of Betti number estimation

We establish the average-case hardness of Betti number estimation on random clique complexes via a reduction from the planted clique problem. We further show that our reduction implies a series of hardness results for many problems in both classical and quantum Topological Data Analysis (qTDA). Under the classical planted clique conjecture, no randomized polynomial-time Betti number estimator achieves additive error below $\tfrac12$ with constant advantage. Under a new quantum planted clique conjecture that we introduce, the same conclusion holds for quantum polynomial-time algorithms. We also obtain related conditional hardness results for homology vanishing, additive approximations with larger error tolerances, preparation of simplex and harmonic states, cycle recovery, and counting eigenvalues at low energy. Our reduction clarifies the structural requirements for quantum advantage in TDA and provides a new lens to investigate the classical and quantum complexity of related problems.

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arXiv2026

Worst-case depth hierarchy for shallow quantum circuits

Circuit depth is a central resource in complexity theory. While bounded-depth classical circuits admit well-understood hierarchy theorems, the internal structure of constant-depth quantum computation remains comparatively unexplored. We prove an explicit depth hierarchy theorem for $\mathsf{QNC}^0$. For each $d\ge 12$, we construct a family of two-round interactive problems on which no depth-$(d-1)$ quantum circuit can achieve near-perfect success, regardless of gate set, circuit size, or ancillary qubits. In contrast, we prove that our construction admits realizations by simple bounded fan-in quantum circuits of depth larger than $d$ by a small constant factor. Moreover, all bounded fan-in classical circuits of sublogarithmic depth (in the input size) fail to achieve perfect success on these tasks for every $d$, yielding a hierarchy of problems that show unconditional quantum advantage of $\mathsf{QNC}^0$ over $\mathsf{NC}^0$. A key obstacle is the scarcity of lower bound techniques for quantum circuits. To address this, we develop methods to analyze how depth affects a circuit's ability to realize nonlocal correlations amongst its output qubits in a fine-grained manner. Our approach exploits the correspondence between constraint systems and nonlocal games, translating group-theoretic constructions into rigid operator-valued constraint systems and then into non-local games. In particular, we construct constraint systems whose unique faithful operator-valued solutions require every perfect strategy, and every near-perfect strategy to a fixed precision, to implement multi-controlled phase operations. This reduces to a nonlocal unitary-synthesis problem, yielding depth lower bounds for both shallow quantum and classical circuits. These results show that increasing depth strictly increases computational power within $\mathsf{QNC}^0$, establishing a genuinely quantum hierarchy.

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arXiv2025

Unconditionally separating noisy $\mathsf{QNC}^0$ from bounded polynomial threshold circuits of constant depth

The rapid evolution of quantum devices fuels concerted efforts to experimentally establish quantum advantage over classical computing. Many demonstrations of quantum advantage, however, rely on computational assumptions and face verification challenges. Furthermore, steady advances in classical algorithms and machine learning make the issue of provable, practically demonstrable quantum advantage a moving target. In this work, we unconditionally demonstrate that parallel quantum computation can exhibit greater computational power than previously recognized. We prove that polynomial-size biased threshold circuits of constant depth -- which model neural networks with tunable expressivity -- fail to solve certain problems solvable by small constant-depth quantum circuits with local gates, for values of the bias that allow quantifiably large computational power. Additionally, we identify a family of problems that are solvable in constant depth by a universal quantum computer over prime-dimensional qudits with bounded connectivity, but remain hard for polynomial-size biased threshold circuits. We thereby bridge the foundational theory of non-local games in higher dimensions with computational advantage on emerging devices operating on a wide range of physical platforms. Finally, we show that these quantum advantages are robust to noise across all prime qudit dimensions with all-to-all connectivity, enhancing their practical appeal.

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arXiv2025

Unconditional Pseudorandomness against Shallow Quantum Circuits

Quantum computational pseudorandomness has emerged as a fundamental notion that spans connections to complexity theory, cryptography and fundamental physics. However, all known constructions of efficient quantum-secure pseudorandom objects rely on complexity theoretic assumptions. In this work, we establish the first unconditionally secure efficient pseudorandom constructions against shallow-depth quantum circuit classes. We prove that: $\bullet$ Any quantum state 2-design yields unconditional pseudorandomness against both $\mathsf{QNC}^0$ circuits with arbitrarily many ancillae and $\mathsf{AC}^0\circ\mathsf{QNC}^0$ circuits with nearly linear ancillae. $\bullet$ Random phased subspace states, where the phases are picked using a 4-wise independent function, are unconditionally pseudoentangled against the above circuit classes. $\bullet$ Any unitary 2-design yields unconditionally secure parallel-query pseudorandom unitaries against geometrically local $\mathsf{QNC}^0$ adversaries, even with limited $\mathsf{AC}^0$ postprocessing. Our indistinguishability results for 2-designs stand in stark contrast to the standard setting of quantum pseudorandomness against $\mathsf{BQP}$ circuits, wherein they can be distinguishable from Haar random ensembles using more than two copies or queries. Our work demonstrates that quantum computational pseudorandomness can be achieved unconditionally for natural classes of restricted adversaries, opening new directions in quantum complexity theory.

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arXiv2025

Quantum Catalytic Space

Space complexity is a key field of study in theoretical computer science. In the quantum setting there are clear motivations to understand the power of space-restricted computation, as qubits are an especially precious and limited resource. Recently, a new branch of space-bounded complexity called catalytic computing has shown that reusing space is a very powerful computational resource, especially for subroutines that incur little to no space overhead. While quantum catalysis in an information theoretic context, and the power of ``dirty'' qubits for quantum computation, has been studied over the years, these models are generally not suitable for use in quantum space-bounded algorithms, as they either rely on specific catalytic states or destroy the memory being borrowed. We define the notion of catalytic computing in the quantum setting and show a number of initial results about the model. First, we show that quantum catalytic logspace can always be computed quantumly in polynomial time; the classical analogue of this is the largest open question in catalytic computing. This also allows quantum catalytic space to be defined in an equivalent way with respect to circuits instead of Turing machines. We also prove that quantum catalytic logspace can simulate log-depth threshold circuits, a class which is known to contain (and believed to strictly contain) quantum logspace, thus showcasing the power of quantum catalytic space. Finally we show that both unitary quantum catalytic logspace and classical catalytic logspace can be simulated in the one-clean qubit model.

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arXiv2025

Do black holes store negative entropy?

The Bekenstein-Hawking equation states that black holes should have entropy proportional to their areas to make black hole physics compatible with the second law of thermodynamics. However, this equation leads to an inconsistency among the first law of black hole mechanics, the entropy conservation law of quantum mechanics, and a heuristic picture for Hawking radiation, creation of entangled pairs near the horizon. Here we propose an equation alternative to the Bekenstein-Hawking equation from the viewpoint of quantum information, to resolve this inconsistency without changing Hawking's original pair-creation picture for the radiation. This argues that the area of any stationary black hole, including Kerr and charged ones, is proportional to the coherent information, which is 'minus' the conditional entropy defined only in the quantum regime, from the outside, to the black hole excluding negative-frequency particles generated by Hawking's pair creation. Our equation suggests that negative-frequency particles inside a black hole behave as if they have `negative' entropy. Our result implies that a black hole stores purely quantum information, rather than classical information, and the area of the event horizon describes the number of Bell pairs that can be distilled between the interior and exterior.

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arXiv2024

Quantum Channel Testing in Average-Case Distance

We study the complexity of testing properties of quantum channels. First, we show that testing identity to any channel $\mathcal N: \mathbb C^{d_{\mathrm{in}} \times d_{\mathrm{in}}} \to \mathbb C^{d_{\mathrm{out}} \times d_{\mathrm{out}}}$ in diamond norm distance requires $Ω(\sqrt{d_{\mathrm{in}}} / \varepsilon)$ queries, even in the strongest algorithmic model that admits ancillae, coherence, and adaptivity. This is due to the worst-case nature of the distance induced by the diamond norm. Motivated by this limitation and other theoretical and practical applications, we introduce an average-case analogue of the diamond norm, which we call the average-case imitation diamond (ACID) norm. In the weakest algorithmic model without ancillae, coherence, or adaptivity, we prove that testing identity to certain types of channels in ACID distance can be done with complexity independent of the dimensions of the channel, while for other types of channels the complexity depends on both the input and output dimensions. Building on previous work, we also show that identity to any fixed channel can be tested with $\tilde O(d_{\mathrm{in}} d_{\mathrm{out}}^{3/2} / \varepsilon^2)$ queries in ACID distance and $\tilde O(d_{\mathrm{in}}^2 d_{\mathrm{out}}^{3/2} / \varepsilon^2)$ queries in diamond distance in this model. Finally, we prove tight bounds on the complexity of channel tomography in ACID distance.

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arXiv2024

Information-theoretic generalization bounds for learning from quantum data

Learning tasks play an increasingly prominent role in quantum information and computation. They range from fundamental problems such as state discrimination and metrology over the framework of quantum probably approximately correct (PAC) learning, to the recently proposed shadow variants of state tomography. However, the many directions of quantum learning theory have so far evolved separately. We propose a general mathematical formalism for describing quantum learning by training on classical-quantum data and then testing how well the learned hypothesis generalizes to new data. In this framework, we prove bounds on the expected generalization error of a quantum learner in terms of classical and quantum information-theoretic quantities measuring how strongly the learner's hypothesis depends on the specific data seen during training. To achieve this, we use tools from quantum optimal transport and quantum concentration inequalities to establish non-commutative versions of decoupling lemmas that underlie recent information-theoretic generalization bounds for classical machine learning. Our framework encompasses and gives intuitively accessible generalization bounds for a variety of quantum learning scenarios such as quantum state discrimination, PAC learning quantum states, quantum parameter estimation, and quantumly PAC learning classical functions. Thereby, our work lays a foundation for a unifying quantum information-theoretic perspective on quantum learning.

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arXiv2023

Quantum Ridgelet Transform: Winning Lottery Ticket of Neural Networks with Quantum Computation

A significant challenge in the field of quantum machine learning (QML) is to establish applications of quantum computation to accelerate common tasks in machine learning such as those for neural networks. Ridgelet transform has been a fundamental mathematical tool in the theoretical studies of neural networks, but the practical applicability of ridgelet transform to conducting learning tasks was limited since its numerical implementation by conventional classical computation requires an exponential runtime $\exp(O(D))$ as data dimension $D$ increases. To address this problem, we develop a quantum ridgelet transform (QRT), which implements the ridgelet transform of a quantum state within a linear runtime $O(D)$ of quantum computation. As an application, we also show that one can use QRT as a fundamental subroutine for QML to efficiently find a sparse trainable subnetwork of large shallow wide neural networks without conducting large-scale optimization of the original network. This application discovers an efficient way in this regime to demonstrate the lottery ticket hypothesis on finding such a sparse trainable neural network. These results open an avenue of QML for accelerating learning tasks with commonly used classical neural networks.

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arXiv2023

A remark on the quantum complexity of the Kronecker coefficients

We prove that the computation of the Kronecker coefficients of the symmetric group is contained in the complexity class #BQP. This improves a recent result of Bravyi, Chowdhury, Gosset, Havlicek, and Zhu. We use only the quantum computing tools that are used in their paper and additional classical representation theoretic insights. We also prove the analogous result for the plethysm coefficients.

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arXiv2022

Quantum Worst-Case to Average-Case Reductions for All Linear Problems

We study the problem of designing worst-case to average-case reductions for quantum algorithms. For all linear problems, we provide an explicit and efficient transformation of quantum algorithms that are only correct on a small (even sub-constant) fraction of their inputs into ones that are correct on all inputs. This stands in contrast to the classical setting, where such results are only known for a small number of specific problems or restricted computational models. En route, we obtain a tight $Ω(n^2)$ lower bound on the average-case quantum query complexity of the Matrix-Vector Multiplication problem. Our techniques strengthen and generalise the recently introduced additive combinatorics framework for classical worst-case to average-case reductions (STOC 2022) to the quantum setting. We rely on quantum singular value transformations to construct quantum algorithms for linear verification in superposition and learning Bogolyubov subspaces from noisy quantum oracles. We use these tools to prove a quantum local correction lemma, which lies at the heart of our reductions, based on a noise-robust probabilistic generalisation of Bogolyubov's lemma from additive combinatorics.

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arXiv2022

Constant-time one-shot testing of large-scale graph states

Fault-tolerant measurement-based quantum computation (MBQC) with recent progress on quantum technologies leads to a promising scalable platform for realizing quantum computation, conducted by preparing a large-scale graph state over many qubits and performing single-qubit measurements on the state. With fault-tolerant MBQC, even if the graph-state preparation suffers from errors occurring at an unknown physical error rate, we can suppress the effect of the errors. Verifying graph states is vital to test whether we can conduct MBQC as desired even with such errors. However, problematically, existing state-of-the-art protocols for graph-state verification by fidelity estimation have required measurements on many copies of the entire graph state and hence have been prohibitively costly in terms of the number of qubits and the runtime. We here construct an efficient alternative framework for testing graph states for fault-tolerant MBQC based on the theory of property testing. Our test protocol accepts with high probability when the physical error rate is small enough to make fault-tolerant MBQC feasible and rejects when the rate is above the threshold of fault-tolerant MBQC. The novelty of our protocol is that we use only a single copy of the $N$-qubit graph state and single-qubit Pauli measurements only on a constant-sized subset of the qubits; thus, the protocol has a constant runtime independently of $N$. Furthermore, we can immediately use the rest of the graph state for fault-tolerant MBQC if the protocol accepts. These results achieve a significant advantage over prior art for graph-state verification in the number of qubits and the total runtime. Consequently, our work offers a new route to a fast and practical framework for benchmarking large-scale quantum state preparation.

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arXiv2021

Sublinear quantum algorithms for estimating von Neumann entropy

Entropy is a fundamental property of both classical and quantum systems, spanning myriad theoretical and practical applications in physics and computer science. We study the problem of obtaining estimates to within a multiplicative factor $γ>1$ of the Shannon entropy of probability distributions and the von Neumann entropy of mixed quantum states. Our main results are: $\quad\bullet$ an $\widetilde{\mathcal{O}}\left( n^{\frac{1+η}{2γ^2}}\right)$-query quantum algorithm that outputs a $γ$-multiplicative approximation of the Shannon entropy $H(\mathbf{p})$ of a classical probability distribution $\mathbf{p} = (p_1,\ldots,p_n)$; $\quad\bullet$ an $\widetilde{\mathcal{O}}\left( n^{\frac12+\frac{1+η}{2γ^2}}\right)$-query quantum algorithm that outputs a $γ$-multiplicative approximation of the von Neumann entropy $S(ρ)$ of a density matrix $ρ\in\mathbb{C}^{n\times n}$. In both cases, the input is assumed to have entropy bounded away from zero by a quantity determined by the parameter $η>0$, since, as we prove, no polynomial query algorithm can multiplicatively approximate the entropy of distributions with arbitrarily low entropy. In addition, we provide $Ω\left(n^{\frac{1}{3γ^2}}\right)$ lower bounds on the query complexity of $γ$-multiplicative estimation of Shannon and von Neumann entropies. We work with the quantum purified query access model, which can handle both classical probability distributions and mixed quantum states, and is the most general input model considered in the literature.

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arXiv2021

Quantum algorithm for estimating Renyi entropies of quantum states

We describe a quantum algorithm to estimate the $α$-Renyi entropy of an unknown density matrix $ρ\in\mathcal{C}^{d\times d}$ for $α\neq 1$ by combining the recent technique of quantum singular value transformations with the method of estimating normalised traces in the one clean qubit model. We consider an oracular input model where the input state is prepared via a quantum oracle that outputs a purified version of the state, assumed to be non-singular. Our method outputs an estimate of the $α$-Renyi entropy to additive precision $ε$, using an expected total number $O\left(\frac{1}{(xε)^2}\right)$ of independent applications of a quantum circuit which coherently queries the input unitary $O\left(\frac{1}δ\log \frac{d}ε\right)$ times, in each case measuring a single output qubit. Here $δ$ is a lower cutoff on the smallest eigenvalue of $ρ$ and $x=\frac{1}{d}\!\mathop{Tr}{ρ^α}$. The expected number of measurements made in this method can be compared to results in the sample complexity model that generally require $Θ(d^2/ε^2)$ samples. Furthermore, we also show that multiplicative approximations can be obtained by iteratively using additive approximations, with an overhead logarithmic in the dimension $d$.

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arXiv2020

Learning with Optimized Random Features: Exponential Speedup by Quantum Machine Learning without Sparsity and Low-Rank Assumptions

Kernel methods augmented with random features give scalable algorithms for learning from big data. But it has been computationally hard to sample random features according to a probability distribution that is optimized for the data, so as to minimize the required number of features for achieving the learning to a desired accuracy. Here, we develop a quantum algorithm for sampling from this optimized distribution over features, in runtime $O(D)$ that is linear in the dimension $D$ of the input data. Our algorithm achieves an exponential speedup in $D$ compared to any known classical algorithm for this sampling task. In contrast to existing quantum machine learning algorithms, our algorithm circumvents sparsity and low-rank assumptions and thus has wide applicability. We also show that the sampled features can be combined with regression by stochastic gradient descent to achieve the learning without canceling out our exponential speedup. Our algorithm based on sampling optimized random features leads to an accelerated framework for machine learning that takes advantage of quantum computers.

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arXiv2020

A Quantum Search Decoder for Natural Language Processing

Probabilistic language models, e.g. those based on an LSTM, often face the problem of finding a high probability prediction from a sequence of random variables over a set of tokens. This is commonly addressed using a form of greedy decoding such as beam search, where a limited number of highest-likelihood paths (the beam width) of the decoder are kept, and at the end the maximum-likelihood path is chosen. In this work, we construct a quantum algorithm to find the globally optimal parse (i.e. for infinite beam width) with high constant success probability. When the input to the decoder is distributed as a power-law with exponent $k>0$, our algorithm has runtime $R^{n f(R,k)}$, where $R$ is the alphabet size, $n$ the input length; here $f<1/2$, and $f\rightarrow 0$ exponentially fast with increasing $k$, hence making our algorithm always more than quadratically faster than its classical counterpart. We further modify our procedure to recover a finite beam width variant, which enables an even stronger empirical speedup while still retaining higher accuracy than possible classically. Finally, we apply this quantum beam search decoder to Mozilla's implementation of Baidu's DeepSpeech neural net, which we show to exhibit such a power law word rank frequency.

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arXiv2019

Spectral sparsification of matrix inputs as a preprocessing step for quantum algorithms

We study the potential utility of classical techniques of spectral sparsification of graphs as a preprocessing step for digital quantum algorithms, in particular, for Hamiltonian simulation. Our results indicate that spectral sparsification of a graph with $n$ nodes through a sampling method, e.g.\ as in \cite{Spielman2011resistances} using effective resistances, gives, with high probability, a locally computable matrix $\tilde H$ with row sparsity at most $\mathcal{O}(\text{poly}\log n)$. For a symmetric matrix $H$ of size $n$ with $m$ non-zero entries, a one-time classical runtime overhead of $\mathcal{O}(m||H||t\log n/ε)$ expended in spectral sparsification is then found to be useful as a way to obtain a sparse matrix $\tilde H$ that can be used to approximate time evolution $e^{itH}$ under the Hamiltonian $H$ to precision $ε$. Once such a sparsifier is obtained, it could be used with a variety of quantum algorithms in the query model that make crucial use of row sparsity. We focus on the case of efficient quantum algorithms for sparse Hamiltonian simulation, since Hamiltonian simulation underlies, as a key subroutine, several quantum algorithms, including quantum phase estimation and recent ones for linear algebra. Finally, we also give two simple quantum algorithms to estimate the row sparsity of an input matrix, which achieve a query complexity of $\mathcal{O}(n^{3/2})$ as opposed to $\mathcal{O}(n^2)$ that would be required by any classical algorithm for the task.

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arXiv2018

Implementing smooth functions of a Hermitian matrix on a quantum computer

We review existing methods for implementing smooth functions f(A) of a sparse Hermitian matrix A on a quantum computer, and analyse a further combination of these techniques which has some advantages of simplicity and resource consumption in some cases. Our construction uses the linear combination of unitaries method with Chebyshev polynomial approximations. The query complexity we obtain is O(log C/eps) where eps is the approximation precision, and C>0 is an upper bound on the magnitudes of the derivatives of the function f over the domain of interest. The success probability depends on the 1-norm of the Taylor series coefficients of f, the sparsity d of the matrix, and inversely on the smallest singular value of the target matrix f(A).

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