arXiv subjects

Noah Linden

Publications and source records attributed to Noah Linden.

At least 19 recordsLinked to original sources

arXiv2026

Reducing measurement error with adaptivity

We show that adaptivity, also called feed-forward, is a powerful resource in reducing the error of measurement circuits that combine multiple uses of a noisy measuring device. In previous work, it has been shown that error in measurements can be mitigated by using the measuring device multiple times. So far, the most effective protocols have been parallel measurement schemes, where all measurements are simultaneous. We extend this idea to adaptive measurement schemes, where the results of previous measurements can influence our choice of processing further down the circuit. We show that adaptive measurement circuits can in general reduce overall measurement errors further than any non-adaptive measurement circuit could with the same number of measurements. In particular, we show that for a large class of noisy two-outcome qubit measurements, such an adaptive advantage can exist when as few as three measurements are used. We also show that the adaptive advantage is unbounded across the class of noisy two-outcome qubit measurements, as the number of uses of the device increases. As part of this work, we devise and use methods for finding optimal measurement circuits in both the non-adaptive and adaptive cases. In addition, we prove general results about the limits of such circuits, both in measuring a qubit, and more generally, a qudit.

quant-ph↗

arXiv2026

Can a spin-half particle ever give more than two spots in a Stern-Gerlach experiment? -- the subtle physics of effective Hamiltonians

We show that a spin-1/2 particle can behave as if it were spin-$s$, and generate $2s+1$ spots in a Stern Gerlach measurement (albeit with a smaller gyromagnetic ratio). This arises from some subtle properties of effective Hamiltonians and Hamiltonians with constraints. Examples of implications of the effect in condensed matter are discussed. We also give some simple non-perturbative bounds for a system subjected to strong constraints.

quant-ph↗

arXiv2026

Automated quantum circuit optimization with randomized replacements

Quantum circuit optimization - the process of transforming a quantum circuit into an equivalent one with reduced time and space requirements - is crucial for maximizing the utility of current and near-future quantum devices. While most automated optimization techniques focus on transforming circuits into equivalent ones that implement the same unitary, we show that substantial new opportunities for resource reduction can be achieved by (1) allowing approximate local transformations and (2) employing mixed quantum channels to approximate pure circuits. Our novel automated protocol for approximate circuit rewriting is a refined evolution of automated optimization techniques based on the ZX-calculus, where we add a greedy strategy that selectively replaces ZX-diagrams with small phase angles with stochastic mixtures of the identity and carefully chosen over-rotations, which are designed to reduce the overall gate count in expectation while staying within a strict error budget. This approach yields modest two-qubit gate count reduction in random quantum circuits, and achieves a substantial reduction in structured circuits such as the quantum Fourier transform. Fundamentally, our protocol converts experimental noise due to gate applications into deliberately engineered random noise, outperforming many other approximation methods on average. These results highlight the potential of mixed-channel approximations to enhance future quantum circuit performance, suggesting new directions for resource-aware automated quantum compilation beyond pure unitary channels.

quant-ph↗

arXiv2025

Adaptive quantum phase estimation can be better than non-adaptive

Quantum phase estimation is one of the most important tools in quantum algorithms. It can be made non-adaptive (meaning all applications of the unitary $U_ϕ$ happen simultaneously) without using more applications of $U_ϕ$, albeit at the expense of using many more qubits. It is also known that there is no advantage for adaptive algorithms in the case where the phase that needs to be estimated is arbitrary or is uniformly random. Here we give examples of a special case of phase estimation, with a promise on the values that the unknown phase can take, where adaptive methods are provably better than non-adaptive methods by a factor of nearly 2 in the number of uses of $U_ϕ$. We also prove some upper bounds on the maximum advantage that adaptive algorithms for phase estimation can achieve over non-adaptive ones.

quant-ph↗

arXiv2025

The Hadamard gate cannot be replaced by a resource state in universal quantum computation

We consider models of quantum computation that involve operations performed on some fixed resourceful quantum state. Examples that fit this paradigm include magic state injection and measurement-based approaches. We introduce a framework that incorporates both of these cases and focus on the role of coherence (or superposition) in this context, as exemplified through the Hadamard gate. We prove that given access to incoherent unitaries (those that are unable to generate superposition from computational basis states, e.g. CNOT, diagonal gates), classical control, computational basis measurements, and any resourceful ancillary state (of arbitrary dimension), it is not possible to implement any coherent unitary (e.g. Hadamard) exactly with non-zero probability. We also consider the approximate case by providing lower bounds for the induced trace distance between the above operations and $n$ Hadamard gates. To demonstrate the stability of this result, this is then extended to a similar no-go result for the case of using $k$ Hadamard gates to exactly implement $n>k$ Hadamard gates.

quant-ph↗

arXiv2025

Fast Scrambling in Classically Simulable Quantum Circuits

We study operator scrambling in quantum circuits built from `super-Clifford' gates. For such circuits it was established in arXiv:2002.12824 that the time evolution of operator entanglement for a large class of many-body operators can be efficiently simulated on a classical computer, including for operators with volume-law entanglement. Here we extend the scope of this formalism in two key ways. Firstly we provide evidence that these classically simulable circuits include examples of fast scramblers, by constructing a circuit for which operator entanglement is numerically found to saturate in a time $t_* \sim \mathrm{ln}(N)$ (with $N$ the number of qubits). Secondly we demonstrate that, in addition to operator entanglement, certain out-of-time ordered correlation functions (OTOCs) can be classically simulated within the same formalism. As a consequence such OTOCs can be computed numerically in super-Clifford circuits with thousands of qubits, and we study several explicit examples in the aforementioned fast scrambling circuits.

quant-ph↗

arXiv2023

How to use arbitrary measuring devices to perform almost perfect measurements

We consider the problem of reproducing one quantum measurement given the ability to perform another. We give a general framework and specific protocols for this problem. For example, we show how to use available "imperfect" devices a small number of times to implement a target measurement with average error that drops off exponentially with the number of imperfect measurements used. We hope that could be useful in near-term applications as a type of lightweight error mitigation of the measuring devices. As well as the view to practical applications, we consider the question from a general theoretical perspective in the most general setting where both the available and target measurements are arbitrary generalised quantum measurements. We show that this general problem in fact reduces to the ability to reproduce the statistics of (complete) von Neumann measurements, and that in the asymptotic limit of infinitely many uses of the available measurement, a simple protocol based upon `classical cloning' can perfectly achieve this task. We show that asymptotically all (non-trivial) quantum measurements are equivalent. We also study optimal protocols for a fixed number of uses of the available measurement. This includes, but is not limited to, improving both noisy and lossy quantum measurements. Furthermore, we show that, in a setting where we perform multiple measurements in parallel, we can achieve finite-rate measurement reproduction, by using block-coding techniques from classical information theory. Finally, we show that advantages can also be gained by making use of probabilistic protocols.

quant-ph↗

arXiv2023

Fidelity estimation of quantum states on a silicon photonic chip

As a measure of the 'closeness' of two quantum states, fidelity plays a fundamental role in quantum information theory. Fidelity estimation protocols try to strike a balance between information gleaned from an experiment, and the efficiency of its implementation, in terms of the number of states consumed by the protocol. Here we adapt a previously reported optimal state verification protocol (Phys. Rev. Lett. 120, 170502, 2018) for fidelity estimation of two-qubit states. We demonstrate the protocol experimentally using a fully-programmable silicon photonic two-qubit chip. Our protocol outputs significantly smaller error bars of its point estimate in comparison with another widely-used estimation protocol, showing a clear step forward in the ability to estimate the fidelity of quantum states produced by a practical device.

quant-ph↗

arXiv2022

Average-Case Verification of the Quantum Fourier Transform Enables Worst-Case Phase Estimation

The quantum Fourier transform (QFT) is a key primitive for quantum computing that is typically used as a subroutine within a larger computation, for instance for phase estimation. As such, we may have little control over the state that is input to the QFT. Thus, in implementing a good QFT, we may imagine that it needs to perform well on arbitrary input states. Verifying this worst-case correct behaviour of a QFT-implementation would be exponentially hard (in the number of qubits) in general, raising the concern that this verification would be impossible in practice on any useful-sized system. In this paper we show that, in fact, we only need to have good average-case performance of the QFT to achieve good worst-case performance for key tasks -- phase estimation, period finding and amplitude estimation. Further we give a very efficient procedure to verify this required average-case behaviour of the QFT.

quant-ph↗

arXiv2022

Quantum majority vote

Majority vote is a basic method for amplifying correct outcomes that is widely used in computer science and beyond. While it can amplify the correctness of a quantum device with classical output, the analogous procedure for quantum output is not known. We introduce quantum majority vote as the following task: given a product state $|ψ_1\rangle \otimes \dots \otimes |ψ_n\rangle$ where each qubit is in one of two orthogonal states $|ψ\rangle$ or $|ψ^\perp\rangle$, output the majority state. We show that an optimal algorithm for this problem achieves worst-case fidelity of $1/2 + Θ(1/\sqrt{n})$. Under the promise that at least $2/3$ of the input qubits are in the majority state, the fidelity increases to $1 - Θ(1/n)$ and approaches $1$ as $n$ increases. We also consider the more general problem of computing any symmetric and equivariant Boolean function $f: \{0,1\}^n \to \{0,1\}$ in an unknown quantum basis, and show that a generalization of our quantum majority vote algorithm is optimal for this task. The optimal parameters for the generalized algorithm and its worst-case fidelity can be determined by a simple linear program of size $O(n)$. The time complexity of the algorithm is $O(n^4 \log n)$ where $n$ is the number of input qubits.

quant-ph↗

arXiv2022

Integrable spin chains and the Clifford group

We construct new families of spin chain Hamiltonians that are local, integrable and translationally invariant. To do so, we make use of the Clifford group that arises in quantum information theory. We consider translation invariant Clifford group transformations that can be described by matrix product operators (MPOs). We classify the translation invariant Clifford group transformations that consist of a shift operator and an MPO of bond dimension two -- this includes transformations that preserve locality of all Hamiltonians; as well as those that lead to non-local images of particular operators but nevertheless preserve locality of certain Hamiltonians. We characterise the translation invariant Clifford group transformations that take single-site Pauli operators to local operators on at most five sites -- examples of Quantum Cellular Automata -- leading to a discrete family of Hamiltonians that are equivalent to the canonical XXZ model under such transformations. For spin chains solvable by algebraic Bethe Ansatz, we explain how conjugating by a matrix product operator affects the underlying integrable structure. This allows us to relate our results to the usual classifications of integrable Hamiltonians. We also treat the case of spin chains solvable by free fermions.

math-ph↗

arXiv2021

Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance

Inspired by recent progress in quantum algorithms for ordinary and partial differential equations, we study quantum algorithms for stochastic differential equations (SDEs). Firstly we provide a quantum algorithm that gives a quadratic speed-up for multilevel Monte Carlo methods in a general setting. As applications, we apply it to compute expectation values determined by classical solutions of SDEs, with improved dependence on precision. We demonstrate the use of this algorithm in a variety of applications arising in mathematical finance, such as the Black-Scholes and Local Volatility models, and Greeks. We also provide a quantum algorithm based on sublinear binomial sampling for the binomial option pricing model with the same improvement.

quant-ph↗

arXiv2021

Lightweight Detection of a Small Number of Large Errors in a Quantum Circuit

Suppose we want to implement a unitary $U$, for instance a circuit for some quantum algorithm. Suppose our actual implementation is a unitary $\tilde{U}$, which we can only apply as a black-box. In general it is an exponentially-hard task to decide whether $\tilde{U}$ equals the intended $U$, or is significantly different in a worst-case norm. In this paper we consider two special cases where relatively efficient and lightweight procedures exist for this task. First, we give an efficient procedure under the assumption that $U$ and $\tilde{U}$ (both of which we can now apply as a black-box) are either equal, or differ significantly in only one $k$-qubit gate, where $k=O(1)$ (the $k$ qubits need not be contiguous). Second, we give an even more lightweight procedure under the assumption that $U$ and $\tilde{U}$ are Clifford circuits which are either equal, or different in arbitrary ways (the specification of $U$ is now classically given while $\tilde{U}$ can still only be applied as a black-box). Both procedures only need to run $\tilde{U}$ a constant number of times to detect a constant error in a worst-case norm. We note that the Clifford result also follows from earlier work of Flammia and Liu, and da Silva, Landon-Cardinal, and Poulin. In the Clifford case, our error-detection procedure also allows us efficiently to learn (and hence correct) $\tilde{U}$ if we have a small list of possible errors that could have happened to $U$; for example if we know that only $O(1)$ of the gates of $\tilde{U}$ are wrong, this list will be polynomially small and we can test each possible erroneous version of $U$ for equality with $\tilde{U}$.

quant-ph↗

arXiv2020

Quantum circuits with classically simulable operator scrambling

We introduce a new family of quantum circuits for which the scrambling of a subspace of non-local operators is classically simulable. We call these circuits `super-Clifford circuits', since the Heisenberg time evolution of these operators corresponds to a Clifford evolution in operator space. By simulating the Clifford evolution in operator space we are able to simulate the time evolution of certain single Pauli strings into operators with an operator entanglement that grows linearly with the number of qubits. These circuits provide a new technique for studying scrambling in systems with a large number of qubits, and are an explicit counter example to the intuition that classical simulability implies the absence of scrambling.

quant-ph↗

arXiv2020

Quantum vs. classical algorithms for solving the heat equation

Quantum computers are predicted to outperform classical ones for solving partial differential equations, perhaps exponentially. Here we consider a prototypical PDE - the heat equation in a rectangular region - and compare in detail the complexities of ten classical and quantum algorithms for solving it, in the sense of approximately computing the amount of heat in a given region. We find that, for spatial dimension $d \ge 2$, there is an at most quadratic quantum speedup using an approach based on applying amplitude estimation to an accelerated classical random walk. However, an alternative approach based on a quantum algorithm for linear equations is never faster than the best classical algorithms.

quant-ph↗

arXiv2020

Quantum speedups of some general-purpose numerical optimisation algorithms

We give quantum speedups of several general-purpose numerical optimisation methods for minimising a function $f:\mathbb{R}^n \to \mathbb{R}$. First, we show that many techniques for global optimisation under a Lipschitz constraint can be accelerated near-quadratically. Second, we show that backtracking line search, an ingredient in quasi-Newton optimisation algorithms, can be accelerated up to quadratically. Third, we show that a component of the Nelder-Mead algorithm can be accelerated by up to a multiplicative factor of $O(\sqrt{n})$. Fourth, we show that a quantum gradient computation algorithm of Gilyén et al. can be used to approximately compute gradients in the framework of stochastic gradient descent. In each case, our results are based on applying existing quantum algorithms to accelerate specific components of the classical algorithms, rather than developing new quantum techniques.

quant-ph↗

arXiv2019

The complexity of compatible measurements

Measurement incompatibility is one of the basic aspects of quantum theory. Here we study the structure of the set of compatible -- i.e. jointly measurable -- measurements. We are interested in whether or not there exist compatible measurements whose parent is maximally complex -- requiring a number of outcomes exponential in the number of measurements, and related questions. Although we show this to be the case in a number of simple scenarios, we show that generically it cannot happen, by proving an upper bound on the number of outcomes of a parent measurement that is linear in the number of compatible measurements. We discuss why this doesn't trivialise the problem of finding parent measurements, but rather shows that a trade-off between memory and time can be achieved. Finally, we also investigate the complexity of extremal compatible measurements in regimes where our bound is not tight, and uncover rich structure.

quant-ph↗

arXiv2019

Structure and efficiency in bacterial photosynthetic light-harvesting

Photosynthetic organisms use networks of chromophores to absorb sunlight and deliver the energy to reaction centres, where charge separation triggers a cascade of chemical steps to store the energy. We present a detailed model of the light-harvesting complexes in purple bacteria, including explicit interaction with sunlight; energy loss through radiative and non-radiative processes; and dephasing and thermalizing effects of coupling to a vibrational bath. An important feature of the model is that we capture the effect of slow vibrational modes by introducing time-dependent disorder. Our model describes the experimentally observed high efficiency of light harvesting, despite the absence of long-range quantum coherence. The one-exciton part of the quantum state fluctuates due to slow vibrational changes, but remains highly mixed at all times. This lack of long-range coherence suggests a relatively minor role for structure in determining the efficiency of bacterial light harvesting. To investigate this we built hypothetical models with randomly arranged chromophores, but still observed high efficiency when typical nearest-neighbour distances are comparable with those found in nature. This helps to explain the efficiency of energy transport in organisms whose chromophore networks differ widely in structure, while also suggesting new design criteria for efficient artificial light-harvesting devices.

physics.bio-ph↗