Abstract

Deep thermalization refers to the emergence of Haar-like randomness from quantum systems upon
partial measurements. As a generalization of quantum thermalization, it is often associated with high
complexity and entanglement. Here, we introduce computational deep thermalization and construct the
fastest possible dynamics exhibiting it at infinite effective temperature. Our circuit dynamics produce
quantum states with low entanglement in polylogarithmic depth that are indistinguishable from Haar random states to any computationally bounded observer. Importantly, the observer is allowed to request
many copies of the same residual state obtained from partial projective measurements on the state--this
condition is beyond the standard settings of quantum pseudorandomness but natural for deep thermalization. In cryptographic terms, these states are pseudorandom and pseudoentangled, and crucially, they
retain these properties under local measurements. Our results demonstrate a new form of computational
thermalization in which thermal-like behavior arises from structured quantum states endowed with
cryptographic properties instead of from highly unstructured ensembles. The low resource complexity of preparing these states suggests scalable simulations of deep thermalization using quantum computers. Our
Letter also motivates the study of computational quantum pseudorandomness beyond BQP observers.

Abstract

Extracting work from a physical system is one of the cornerstones of quantum thermodynamics. The extractable work, as quantified by ergotropy, necessitates a complete description of the quantum system. This is significantly more challenging when the state of the underlying system is unknown, as quantum tomography is extremely inefficient. In this article, we analyze the number of samples of the unknown state required to extract work. With only a single copy of an unknown state, we prove that ergotropy approaches zero in the asymptotic limit, rendering work extraction nearly impossible. In contrast, when multiple copies are available, we quantify the sample complexity required to estimate extractable work, establishing a scaling relationship that balances the desired accuracy with success probability. Our work develops a sample-efficient protocol to assess the utility of unknown states as quantum batteries and opens avenues for estimating thermodynamic quantities using near-term quantum computers.

Abstract

We develop randomized quantum algorithms to simulate quantum collision models, also known as repeated interaction schemes, which provide a rich framework to model various open-system dynamics. The underlying technique involves composing time evolutions of the total (system, bath, and interaction) Hamiltonian and intermittent tracing out the environment degrees of freedom. This results in a unified framework where any near-term Hamiltonian simulation algorithm can be incorporated to implement an arbitrary number of such collisions on early fault-tolerant quantum computers: we do not assume access to specialized oracles such as block encodings and minimize the number of ancilla qubits needed. In particular, using the correspondence between Lindbladian evolution and completely positive trace-preserving maps arising out of memoryless collisions, we provide an end-to-end quantum algorithm for simulating Lindbladian dynamics. For a system of 𝑛-qubits, we exhaustively compare the circuit depth needed to estimate the expectation value of an observable with respect to the reduced state of the system after time 𝑡 while employing different near-term Hamiltonian simulation techniques, requiring at most 𝑛+2 qubits in all. We compare the cnot gate counts of the various approaches for estimating the transverse field magnetization of a 10-qubit XX-Heisenberg spin chain under amplitude damping. Finally, we also develop a framework to efficiently simulate an arbitrary number of memory-retaining collisions, i.e., where environments interact, leading to non-Markovian dynamics. Overall, our methods can leverage quantum collision models for both Markovian and non-Markovian dynamics on early fault-tolerant quantum computers, shedding light on the advantages and limitations of simulating open systems dynamics using this framework.

Abstract

In the circuit model of quantum computing, amplitude amplification techniques can be used to find solutions to NP-hard problems defined on n-bits in time poly(n)2^{n/2}. In this work, we investigate whether such general statements can be made for adiabatic quantum optimization, as provable results regarding its performance are mostly unknown. Although a lower bound of Ω(2^{n/2})has existed in such a setting for over a decade, a purely adiabatic algorithm with this running time has been absent. We show that adiabatic quantum optimization using an unstructured search approach results in a running time that matches this lower bound (up to a polylogarithmic factor) for a broad class of classical local spin Hamiltonians. For this, it is necessary to bound the spectral gap throughout the adiabatic evolution and compute beforehand the position of the avoided crossing with sufficient precision so as to adapt the adiabatic schedule accordingly. However, we show that the position of the avoided crossing is approximately given by a quantity that depends on the degeneracies and inverse gaps of the problem Hamiltonian and is NP-hard to compute even within a low additive precision. Furthermore, computing it exactly (or nearly exactly) is #P-hard. Our work indicates a possible limitation of adiabatic quantum optimization algorithms, leaving open the question of whether provable Grover-like speed-ups can be obtained for any optimization problem using this approach.

Abstract

We develop new algorithms for Quantum Singular Value Transformation (QSVT), a unifying framework underlying a wide range of quantum algorithms. Existing implementations of QSVT rely on block encoding, incurring $O(log L)$ ancilla overhead and circuit depth $widetilde{O}(d lambda L)$ for polynomial transformations of a Hamiltonian $H = sum_{k=1}^L lambda_k H_k$, where $d$ is the polynomial degree, and $lambda = sum_k |lambda_k|$.

We introduce a new approach that eliminates block encoding, needs only a single ancilla qubit, and maintains near-optimal complexity, using only basic Hamiltonian simulation methods such as Trotterization. Our method achieves a circuit depth of $widetilde{O}left(L (d lambda_{text{comm}})^{1+o(1)}right)$, without any multi-qubit controlled gates. Here, $lambda_{text{comm}}$ depends on the nested commutators of the $H_k$'s and can be much smaller than $lambda$.

Central to our technique is a novel use of Richardson extrapolation, enabling systematic error cancellation in interleaved sequences of arbitrary unitaries and Hamiltonian evolution operators, establishing a broadly applicable framework beyond QSVT.

Additionally, we propose two randomized QSVT algorithms for cases with only sampling access to Hamiltonian terms. The first uses qDRIFT, while the second replaces block encodings in QSVT with randomly sampled unitaries. Both achieve quadratic complexity in $d$, which we establish as a lower bound for any randomized method implementing polynomial transformations in this model.

Finally, as applications, we develop end-to-end quantum algorithms for quantum linear systems and ground state property estimation, achieving near-optimal complexity without oracular access. Our results provide a new framework for quantum algorithms, reducing hardware overhead while maintaining near-optimal performance, with implications for both near-term and fault-tolerant quantum computing.

Abstract

In the circuit model of quantum computing, amplitude amplification techniques can be used to find solutions to NP-hard problems defined on n-bits in time poly(n)2^{n/2}. In this work, we investigate whether such general statements can be made for adiabatic quantum optimization, as provable results regarding its performance are mostly unknown. Although a lower bound of Ω(2^{n/2}) has existed in such a setting for over a decade, a purely adiabatic algorithm with this running time has been absent. We show that adiabatic quantum optimization using an unstructured search approach results in a running time that matches this lower bound (up to a polylogarithmic factor) for a broad class of classical local spin Hamiltonians. For this, it is necessary to bound the spectral gap throughout the adiabatic evolution and compute beforehand the position of the avoided crossing with sufficient precision so as to adapt the adiabatic schedule accordingly. However, we show that the position of the avoided crossing is approximately given by a quantity that depends on the degeneracies and inverse gaps of the problem Hamiltonian and is NP-hard to compute even within a low additive precision. Furthermore, computing it exactly (or nearly exactly) is #P-hard. Our work indicates a possible limitation of adiabatic quantum optimization algorithms, leaving open the question of whether provable Grover-like speed-ups can be obtained for any optimization problem using this approach.

Abstract

We develop three new methods to implement any Linear Combination of Unitaries (LCU), a powerful quantum algorithmic tool with diverse applications. While the standard LCU procedure requires several ancilla qubits and sophisticated multi-qubit controlled operations, our methods consume significantly fewer quantum resources. The first method (Single-Ancilla LCU) estimates expectation values of observables with respect to any quantum state prepared by an LCU procedure while requiring only a single ancilla qubit, and no multi-qubit controlled operations. The second approach (Analog LCU) is a simple, physically motivated, continuous-time analogue of LCU, tailored to hybrid qubit-qumode systems. The third method (Ancilla-free LCU) requires no ancilla qubit at all and is useful when we are interested in the projection of a quantum state (prepared by the LCU procedure) in some subspace of interest. We apply the first two techniques to develop new quantum algorithms for a wide range of practical problems, ranging from Hamiltonian simulation, ground state preparation and property estimation, and quantum linear systems. Remarkably, despite consuming fewer quantum resources they retain a provable quantum advantage. The third technique allows us to connect discrete and continuous-time quantum walks with their classical counterparts. It also unifies the recently developed optimal quantum spatial search algorithms in both these frameworks, and leads to the development of new ones that require fewer ancilla qubits. Overall, our results are quite generic and can be readily applied to other problems, even beyond those considered here.

Abstract

Over the years, the framework of Linear combination of unitaries (LCU) has been extremely useful for designing a plethora of quantum algorithms. In this work, we explore whether this widely applicable paradigm can be implemented on quantum computers that will be available immediately after the current NISQ stage. To this end, we develop three variants of LCU and apply each, to quantum algorithms of practical interest. First, we develop a physically motivated, continuous-time analogue of LCU ("Analog LCU"). This technique, implementable on hybrid qubit-qumode systems, is simpler than its discrete-time counterpart. We use this method to develop analog quantum algorithms for ground state preparation and quantum linear systems. We also develop a randomized quantum algorithm to sample from functions of Hamiltonians applied to quantum states ("Single-Ancilla LCU"). This approach repeatedly samples from a short-depth quantum circuit and uses only a single ancilla qubit. We use this to estimate expectation values of observables in the ground states of a Hamiltonian, and in the solution of quantum linear systems. This method is suitable for early fault-tolerant quantum computers. Our third approach stems from the observation that for several applications, it suffices to replace LCU with randomized sampling of unitaries according to the distribution of the LCU coefficients ("Ancilla-free LCU"). This is particularly useful when one is interested in the projection of a quantum state implemented by an LCU procedure in some subspace of interest. We demonstrate that this technique applies to the spatial search problem and helps establish a relationship between discrete and continuous-time quantum walks with their classical counterparts. Our work demonstrates that generic quantum algorithmic paradigms, such as LCU, can potentially be implemented on intermediate-term quantum devices.

Abstract

Linear regression is a widely used technique to fit linear models and finds widespread applications across different areas such as machine learning and statistics. In most real-world scenarios, however, linear regression problems are often ill-posed or the underlying model suffers from overfitting, leading to erroneous or trivial solutions. This is often dealt with by adding extra constraints, known as regularization. In this paper, we use the frameworks of block-encoding and quantum singular value transformation (QSVT) to design the first quantum algorithms for quantum least squares with general ℓ2-regularization. These include regularized versions of quantum ordinary least squares, quantum weighted least squares, and quantum generalized least squares. Our quantum algorithms substantially improve upon prior results on quantum ridge regression (polynomial improvement in the condition number and an exponential improvement in accuracy), which is a particular case of our result. To this end, we assume approximate block-encodings of the underlying matrices as input and use robust QSVT algorithms for various linear algebra operations. In particular, we develop a variable-time quantum algorithm for matrix inversion using QSVT, where we use quantum eigenvalue discrimination as a subroutine instead of gapped phase estimation. This ensures that substantially fewer ancilla qubits are required for this procedure than prior results. Owing to the generality of the block-encoding framework, our algorithms are applicable to a variety of input models and can also be seen as improved and generalized versions of prior results on standard (non-regularized) quantum least squares algorithms.

Abstract

In this article, we undertake a detailed study of the limiting behavior of a three-state discrete-time quantum walk on a one-dimensional lattice with generalized Grover coins. Two limit theorems are proved and consequently we show that the quantum walk exhibits localization at its initial position for a wide range of coin parameters. Finally, we discuss the effect of the coin parameters on the peak velocities of probability distributions of the underlying quantum walks.