arXiv subjects

Peter S. Turner

Publications and source records attributed to Peter S. Turner.

At least 19 recordsLinked to original sources

arXiv2025

Quantum circuits for simulating linear interferometers

Motivated by recent proposals for quantum proof of work protocols, we investigate approaches for simulating linear optical interferometers using digital quantum circuits. We focus on a second quantisation approach, where the quantum computer's registers represent optical modes. We can then use standard quantum optical techniques to decompose the unitary matrix describing an interferometer into an array of 2 by 2 unitaries, which are subsequently synthesised into quantum circuits and stitched together to complete the circuit. For an $m$ mode interferometer with $n$ identical photons, this method requires approximately $\mathcal{O}(m \log(n))$ qubits and a circuit depth of $\mathcal{O}(m n^4 \log_2(n) \: \textrm{polylog}(n^4 / ε))$. We present a software package Aquinas (A quantum interferometer assembler) that uses this approach to generate such quantum circuits. For reference, an arbitrary five mode interferometer with two identical photons is compiled to a 10 qubit quantum circuit with a depth of 1972.

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arXiv2019

Quantum Absorbance Estimation and the Beer-Lambert Law

The utility of transmission measurement has made it a target for quantum enhanced measurement strategies. Here we find if the length of an absorbing object is a controllable variable, then via the Beer-Lambert law, classical strategies can be optimised to reach within 83% of the absolute quantum limit. Our analysis includes experimental losses, detector noise, and input states with arbitrary photon statistics. We derive optimal operating conditions for both classical and quantum sources, and observe experimental agreement with theory using Fock and thermal states.

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arXiv2019

Classically simulating near-term partially-distinguishable and lossy boson sampling

Boson Sampling is the problem of sampling from the same distribution as indistinguishable single photons at the output of a linear optical interferometer. It is an example of a non-universal quantum computation which is believed to be feasible in the near term and cannot be simulated on a classical machine. Like all purported demonstrations of "quantum supremacy", this motivates optimizing classical simulation schemes for a realistic model of the problem, in this case Boson Sampling when the implementations experience lost or distinguishable photons. Although current simulation schemes for sufficiently imperfect boson sampling are classically efficient, in principle the polynomial runtime can be infeasibly large. In this work, we develop a scheme for classical simulation of Boson Sampling under uniform distinguishability and loss, based on the idea of sampling from distributions where at most k photons are indistinguishable. We show that asymptotically this scheme can provide a polynomial improvement in the runtime compared to classically simulating idealised Boson Sampling. More significantly, we show that in the regime considered experimentally relevant, our approach gives an substantial improvement in runtime over other classical simulation approaches.

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arXiv2018

Precision-guaranteed quantum tomography

Quantum state tomography is the standard tool in current experiments for verifying that a state prepared in the lab is close to an ideal target state, but up to now there were no rigorous methods for evaluating the precision of the state preparation in tomographic experiments. We propose a new estimator for quantum state tomography, and prove that the (always physical) estimates will be close to the true prepared state with high probability. We derive an explicit formula for evaluating how high the probability is for an arbitrary finite-dimensional system and explicitly give the one- and two-qubit cases as examples. This formula applies for any informationally complete sets of measurements, arbitrary finite number of data sets, and general loss functions including the infidelity, the Hilbert-Schmidt, and the trace distances. Using the formula, we can evaluate not only the difference between the estimated and prepared states, but also the difference between the prepared and target states. This is the first result directly applicable to the problem of evaluating the precision of estimation and preparation in quantum tomographic experiments.

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arXiv2018

Discriminating distinguishability

Particle distinguishability is a significant challenge for quantum technologies, in particular photonics where the Hong-Ou-Mandel (HOM) effect clearly demonstrates it is detrimental to quantum interference. We take a representation theoretic approach in first quantisation, separating particles' Hilbert spaces into degrees of freedom that we control and those we do not, yielding a quantum information inspired bipartite model where distinguishability can arise as correlation with an environment carried by the particles themselves. This makes clear that the HOM experiment is an instance of a (mixed) state discrimination protocol, which can be generalised to interferometers that discriminate unambiguously between ideal indistinguishable states and interesting distinguishable states, leading to bounds on the success probability of an arbitrary HOM generalisation for multiple particles and modes. After setting out the first quantised formalism in detail, we consider several scenarios and provide a combination of analytical and numerical results for up to nine photons in nine modes. Although the Quantum Fourier Transform features prominently, we see that it is suboptimal for discriminating completely distinguishable states.

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arXiv2018

Quantum simulation of partially distinguishable boson sampling

Boson Sampling is the problem of sampling from the same output probability distribution as a collection of indistinguishable single photons input into a linear interferometer. It has been shown that, subject to certain computational complexity conjectures, in general the problem is difficult to solve classically, motivating optical experiments aimed at demonstrating quantum computational "supremacy". There are a number of challenges faced by such experiments, including the generation of indistinguishable single photons. We provide a quantum circuit that simulates bosonic sampling with arbitrarily distinguishable particles. This makes clear how distinguishabililty leads to decoherence in the standard quantum circuit model, allowing insight to be gained. At the heart of the circuit is the quantum Schur transform, which follows from a representation theoretic approach to the physics of distinguishable particles in first quantisation. The techniques are quite general and have application beyond boson sampling.

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arXiv2017

Generating entanglement with linear optics

Entanglement is the basic building block of linear optical quantum computation, and as such understanding how to generate it in detail is of great importance for optical architectures. We prove that Bell states cannot be generated using only 3 photons in the dual-rail encoding, and give strong numerical evidence for the optimality of the existing 4 photon schemes. In a setup with a single photon in each input mode, we find a fundamental limit on the possible entanglement between a single mode Alice and arbitrary Bob. We investigate and compare other setups aimed at characterizing entanglement in settings more general than dual-rail encoding. The results draw attention to the trade-off between the entanglement a state has and the probability of postselecting that state, which can give surprising constant bounds on entanglement even with increasing numbers of photons.

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arXiv2016

Randomized benchmarking in measurement-based quantum computing

Randomized benchmarking is routinely used as an efficient method for characterizing the performance of sets of elementary logic gates in small quantum devices. In the measurement-based model of quantum computation, logic gates are implemented via single-site measurements on a fixed universal resource state. Here we adapt the randomized benchmarking protocol for a single qubit to a linear cluster state computation, which provides partial, yet efficient characterization of the noise associated with the target gate set. Applying randomized benchmarking to measurement-based quantum computation exhibits an interesting interplay between the inherent randomness associated with logic gates in the measurement-based model and the random gate sequences used in benchmarking. We consider two different approaches: the first makes use of the standard single-qubit Clifford group, while the second uses recently introduced (non-Clifford) measurement-based 2-designs, which harness inherent randomness to implement gate sequences.

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arXiv2016

Postselective quantum interference of distinguishable particles

We show that it is possible for completely distinguishable particles to interfere postselectively without operating on, or indeed having any knowledge of, the distinguishing degree of freedom. In particular, we find a family of three-mode spatial interferometers that, upon inputting, say, a red photon in port 1 and a green photon in port 2, produce a state such that when vacuum is detected at one output, the two photons in the other two outputs will pass a HOM test, despite their frequency degree of freedom remaining untouched. In doing so we develop a general approach to distinguishability based on the Schmidt decomposition between particles' "System" and "Label" degrees of freedom, corresponding to what has been called unitary-unitary duality in many-body physics. This also gives insight into the relationship between the first and second quantized pictures, which is useful in bringing other quantum information concepts across from the former to the latter.

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arXiv2015

Derandomizing quantum circuits with measurement based unitary designs

Entangled multipartite states are resources for universal quantum computation, but they can also give rise to ensembles of unitary transformations, a topic usually studied in the context of random quantum circuits. Using several graph state techniques, we show that these resources can `derandomize' circuit results by sampling the same kinds of ensembles quantum mechanically, (analogously to a quantum random number generator). Furthermore, we find simple examples that give rise to new ensembles whose statistical moments exactly match those of the uniformly random distribution over all unitaries up to order $t$, while foregoing adaptive feed-forward entirely. Such ensembles -- known as $t$-designs -- often cannot be distinguished from the `truly' random ensemble, and so they find use in many applications that require this implied notion of pseudorandomness.

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arXiv2014

Quantum Data Compression of a Qubit Ensemble

Data compression is a ubiquitous aspect of modern information technology, and the advent of quantum information raises the question of what types of compression are feasible for quantum data, where it is especially relevant given the extreme difficulty involved in creating reliable quantum memories. We present a protocol in which an ensemble of quantum bits (qubits) can in principle be perfectly compressed into exponentially fewer qubits. We then experimentally implement our algorithm, compressing three photonic qubits into two. This protocol sheds light on the subtle differences between quantum and classical information. Furthermore, since data compression stores all of the available information about the quantum state in fewer physical qubits, it could provide a vast reduction in the amount of quantum memory required to store a quantum ensemble, making even today's limited quantum memories far more powerful than previously recognized.

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arXiv2013

Testing randomness with photons

Generating and characterising randomness is fundamentally important in both classical and quantum information science. Here we report the experimental demonstration of ensembles of pseudorandom optical processes comprising what are known as t-designs. We show that in practical scenarios, certain finite ensembles of two-mode transformations---1- and 2-designs---are indistinguishable from truly random operations for 1- and 2-photon quantum interference, but they fail to mimic randomness for 2- and 3-photon cases, respectively. This provides a novel optical test of pseudo randomness. We make use of the fact that t-photon behaviour is governed by degree-2t polynomials in the parameters of the optical process to experimentally verify the ensembles' behaviour for complete bases of polynomials. This ensures that outputs will be uniform for arbitrary configurations, satisfying the strict definition of pseudorandomness implicit in the mathematical definition.

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arXiv2012

Effect of nonnegativity on estimation errors in one-qubit state tomography with finite data

We analyze the behavior of estimation errors evaluated by two loss functions, the Hilbert-Schmidt distance and infidelity, in one-qubit state tomography with finite data. We show numerically that there can be a large gap between the estimation errors and those predicted by an asymptotic analysis. The origin of this discrepancy is the existence of the boundary in the state space imposed by the requirement that density matrices be nonnegative (positive semidefinite). We derive an explicit form of a function reproducing the behavior of the estimation errors with high accuracy by introducing two approximations: a Gaussian approximation of the multinomial distributions of outcomes, and linearizing the boundary. This function gives us an intuition for the behavior of the expected losses for finite data sets. We show that this function can be used to determine the amount of data necessary for the estimation to be treated reliably with the asymptotic theory. We give an explicit expression for this amount, which exhibits strong sensitivity to the true quantum state as well as the choice of measurement.

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arXiv2012

Phase-random states: ensembles of states with fixed amplitudes and uniformly distributed phases in a fixed basis

Motivated by studies of typical properties of quantum states in statistical mechanics, we introduce phase-random states, an ensemble of pure states with fixed amplitudes and uniformly distributed phases in a fixed basis. We first show that canonical states typically appear in subsystems of phase-random states. We then investigate the simulatability of phase-random states, which is directly related to that of time evolution in closed systems, by studying their entanglement properties. We find that starting from a separable state, time evolutions under Hamiltonians composed of only separable eigenstates generate extremely high entanglement and are difficult to simulate with matrix product states. We also show that random quantum circuits consisting of only two-qubit diagonal unitaries can generate an ensemble with the same average entanglement as phase-random states.

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arXiv2012

Adaptive experimental design for one-qubit state estimation with finite data based on a statistical update criterion

We consider 1-qubit mixed quantum state estimation by adaptively updating measurements according to previously obtained outcomes and measurement settings. Updates are determined by the average-variance-optimality (A-optimality) criterion, known in the classical theory of experimental design and applied here to quantum state estimation. In general, A-optimization is a nonlinear minimization problem; however, we find an analytic solution for 1-qubit state estimation using projective measurements, reducing computational effort. We compare numerically two adaptive and two nonadaptive schemes for finite data sets and show that the A-optimality criterion gives more precise estimates than standard quantum tomography.

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arXiv2011

Entanglement cost of implementing controlled-unitary operations

We investigate the minimum entanglement cost of the deterministic implementation of two-qubit controlled-unitary operations using local operations and classical communication (LOCC). We show that any such operation can be implemented by a three-turn LOCC protocol, which requires at least 1 ebit of entanglement when the resource is given by a bipartite entangled state with Schmidt number 2. Our result implies that there is a gap between the minimum entanglement cost and the entangling power of controlled-unitary operations. This gap arises due to the requirement of implementing the operations while oblivious to the identity of the inputs.

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arXiv2011

The curious nonexistence of Gaussian 2-designs

2-designs -- ensembles of quantum pure states whose 2nd moments equal those of the uniform Haar ensemble -- are optimal solutions for several tasks in quantum information science, especially state and process tomography. We show that Gaussian states cannot form a 2-design for the continuous-variable (quantum optical) Hilbert space L2(R). This is surprising because the affine symplectic group HWSp (the natural symmetry group of Gaussian states) is irreducible on the symmetric subspace of two copies. In finite dimensional Hilbert spaces, irreducibility guarantees that HWSp-covariant ensembles (such as mutually unbiased bases in prime dimensions) are always 2-designs. This property is violated by continuous variables, for a subtle reason: the (well-defined) HWSp-invariant ensemble of Gaussian states does not have an average state because the averaging integral does not converge. In fact, no Gaussian ensemble is even close (in a precise sense) to being a 2-design. This surprising difference between discrete and continuous quantum mechanics has important implications for optical state and process tomography.

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arXiv2011

Quantum computation over the butterfly network

In order to investigate distributed quantum computation under restricted network resources, we introduce a quantum computation task over the butterfly network where both quantum and classical communications are limited. We consider deterministically performing a two-qubit global unitary operation on two unknown inputs given at different nodes, with outputs at two distinct nodes. By using a particular resource setting introduced by M. Hayashi [Phys. Rev. A \textbf{76}, 040301(R) (2007)], which is capable of performing a swap operation by adding two maximally entangled qubits (ebits) between the two input nodes, we show that unitary operations can be performed without adding any entanglement resource, if and only if the unitary operations are locally unitary equivalent to controlled unitary operations. Our protocol is optimal in the sense that the unitary operations cannot be implemented if we relax the specifications of any of the channels. We also construct protocols for performing controlled traceless unitary operations with a 1-ebit resource and for performing global Clifford operations with a 2-ebit resource.

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