arXiv subjects

Filippo M. Miatto

Publications and source records attributed to Filippo M. Miatto.

At least 19 recordsLinked to original sources

arXiv2023

A Quadratic Speedup in the Optimization of Noisy Quantum Optical Circuits

Linear optical quantum circuits with photon number resolving (PNR) detectors are used for both Gaussian Boson Sampling (GBS) and for the preparation of non-Gaussian states such as Gottesman-Kitaev-Preskill (GKP), cat and NOON states. They are crucial in many schemes of quantum computing and quantum metrology. Classically optimizing circuits with PNR detectors is challenging due to their exponentially large Hilbert space, and quadratically more challenging in the presence of decoherence as state vectors are replaced by density matrices. To tackle this problem, we introduce a family of algorithms that calculate detection probabilities, conditional states (as well as their gradients with respect to circuit parametrizations) with a complexity that is comparable to the noiseless case. As a consequence we can simulate and optimize circuits with twice the number of modes as we could before, using the same resources. More precisely, for an $M$-mode noisy circuit with detected modes $D$ and undetected modes $U$, the complexity of our algorithm is $O(M^2 \prod_{i\in U} C_i^2 \prod_{i\in D} C_i)$, rather than $O(M^2 \prod_{i \in D\cup U} C_i^2)$, where $C_i$ is the Fock cutoff of mode $i$. As a particular case, our approach offers a full quadratic speedup for calculating detection probabilities, as in that case all modes are detected. Finally, these algorithms are implemented and ready to use in the open-source photonic optimization library MrMustard.

quant-ph↗

arXiv2021

Natural Gradient Optimization for Optical Quantum Circuits

Optical quantum circuits can be optimized using gradient descent methods, as the gates in a circuit can be parametrized by continuous parameters. However, the parameter space as seen by the cost function is not Euclidean, which means that the Euclidean gradient does not generally point in the direction of steepest ascent. In order to retrieve the steepest ascent direction, in this work we implement Natural Gradient descent in the optical quantum circuit setting, which takes the local metric tensor into account. In particular, we adapt the Natural Gradient approach to a complex-valued parameter space. We then compare the Natural Gradient approach to vanilla gradient descent and to Adam over two state preparation tasks: a single-photon source and a Gottesman-Kitaev-Preskill state source. We observe that the NG approach has a faster convergence (due in part to the possibility of using larger learning rates) and a significantly smoother decay of the cost function throughout the optimization.

quant-ph↗

arXiv2021

Fast differentiable evolution of quantum states under Gaussian transformations

In a recent work we presented a recursive algorithm to compute the matrix elements of a generic Gaussian transformation in the photon-number basis. Its purpose was to evolve a quantum state by building the transformation matrix and subsequently computing the matrix-vector product. Here we present a faster algorithm that computes the final state without having to generate the full transformation matrix first. With this algorithm we bring the time complexity of computing the Gaussian evolution of an $N$-dimensional $M$-mode state from $O(MN^{2M})$ to $O(M(N^2/2)^M)$, which is an exponential improvement in the number of modes. In the special case of high squeezing, the evolved state can be approximated with complexity $O(MN^{M})$. Our new algorithm is differentiable, which means we can use it in conjunction with gradient-based optimizers for circuit optimization tasks. We benchmark our algorithm by optimizing circuits to produce single photons, Gottesman-Kitaev-Preskill states and NOON states, showing that it is up to one order of magnitude faster than the state of the art.

quant-ph↗

arXiv2020

Fast optimization of parametrized quantum optical circuits

Parametrized quantum optical circuits are a class of quantum circuits in which the carriers of quantum information are photons and the gates are optical transformations. Classically optimizing these circuits is challenging due to the infinite dimensionality of the photon number vector space that is associated to each optical mode. Truncating the space dimension is unavoidable, and it can lead to incorrect results if the gates populate photon number states beyond the cutoff. To tackle this issue, we present an algorithm that is orders of magnitude faster than the current state of the art, to recursively compute the exact matrix elements of Gaussian operators and their gradient with respect to a parametrization. These operators, when augmented with a non-Gaussian transformation such as the Kerr gate, achieve universal quantum computation. Our approach brings two advantages: first, by computing the matrix elements of Gaussian operators directly, we don't need to construct them by combining several other operators; second, we can use any variant of the gradient descent algorithm by plugging our gradients into an automatic differentiation framework such as TensorFlow or PyTorch. Our results will find applications in quantum optical hardware research, quantum machine learning, optical data processing, device discovery and device design.

quant-ph↗

arXiv2020

Hong-Ou-Mandel interference of unconventional temporal laser modes

The Hong-Ou-Mandel (HOM) effect ranks among the most notable quantum interference phenomena, and is central to many applications in quantum technologies. The fundamental effect appears when two independent and indistinguishable photons are superimposed on a beam splitter, which achieves a complete suppression of coincidences between the two output ports. Much less studied, however, is when the fields share coherence (continuous-wave lasers) or mode envelope properties (pulsed lasers). In this case, we expect the existence of two distinct and concurrent HOM interference regimes: the traditional HOM dip on the coherence length time scale, and a structured HOM interference pattern on the pulse length scale. We develop a theoretical framework that describes HOM interference for laser fields having arbitrary temporal waveforms and only partial overlap in time. We observe structured HOM interference from a continuous-wave laser via fast polarization modulation and time-resolved single photon detection fast enough to resolve these structured HOM dips.

quant-ph↗

arXiv2020

Two-photon interference: the Hong-Ou-Mandel effect

Nearly 30 years ago, two-photon interference was observed, marking the beginning of a new quantum era. Indeed, two-photon interference has no classical analogue, giving it a distinct advantage for a range of applications. The peculiarities of quantum physics may now be used to our advantage to outperform classical computations, securely communicate information, simulate highly complex physical systems and increase the sensitivity of precise measurements. This separation from classical to quantum physics has motivated physicists to study two-particle interference for both fermionic and bosonic quantum objects. So far, two-particle interference has been observed with massive particles, among others, such as electrons and atoms, in addition to plasmons, demonstrating the extent of this effect to larger and more complex quantum systems. A wide array of novel applications to this quantum effect is to be expected in the future. This review will thus cover the progress and applications of two-photon (two-particle) interference over the last three decades.

quant-ph↗

arXiv2020

Growing Random Graphs with Quantum Rules

Random graphs are a central element of the study of complex dynamical networks such as the internet, the brain, or socioeconomic phenomena. New methods to generate random graphs can spawn new applications and give insights into more established techniques. We propose two variations of a model to grow random graphs and trees, based on continuous-time quantum walks on the graphs. After a random characteristic time, the position of the walker(s) is measured and new nodes are attached to the nodes where the walkers collapsed. Such dynamical systems are reminiscent of the class of spontaneous collapse theories in quantum mechanics. We investigate several rates of this spontaneous collapse for an individual quantum walker and for two non-interacting walkers. We conjecture (and report some numerical evidence) that the models are scale-free.

quant-ph↗

arXiv2019

Recursive multivariate derivatives of $e^{f(x_1,\dots, x_n)}$ of arbitrary order

High-order derivatives of nested functions of a single variable can be computed with the celebrated Faà di Bruno's formula. Although generalizations of such formula to multiple variables exist, their combinatorial nature generates an explosion of factors, and when the order of the derivatives is high, it becomes very challenging to compute them. A solution is to reuse what has already been computed, which is a built-in feature of recursive implementations. Thanks to this, recursive formulas can play an important role in Machine Learning applications, in particular for Automatic Differentiation. In this manuscript we provide a recursive formula to compute multivariate derivatives of arbitrary order of $e^{f(x_1,\dots,x_n)}$ with respect to the variables $x_i$. We note that this method could also be beneficial in cases where the high-order derivatives of a function $f(x_1,\dots,x_n)$ are hard to compute, but where the derivatives of $\log(f(x_1,\dots,x_n))$ are simpler.

math.NA↗

arXiv2019

Graphical Calculus for products and convolutions

Graphical calculus is an intuitive visual notation for manipulating tensors and index contractions. Using graphical calculus leads to simple and memorable derivations, and with a bit of practice one can learn to prove complex identities even without the need for pen and paper. This manuscript is meant as a demonstration of the power and flexibility of graphical notation and we advocate exploring the use of graphical calculus in undergraduate courses. In the first part we define the following matrix products in graphical language: dot, tensor, Kronecker, Hadamard, Kathri-Rao and Tracy-Singh. We then use our definitions to prove several known identities in an entirely graphical way. Despite ordinary proofs consist in several lines of quite involved mathematical expressions, graphical calculus is so expressive that after writing an identity in graphical form we can realise by visual inspection that it is in fact true. As an example of the intuitiveness of graphical proofs, we derive two new identities. In the second part we develop a graphical description of convolutions, which is a central ingredient of convolutional neural networks and signal processing. Our single definition includes as special cases the circular discrete convolution and the cross-correlation. We illustrate how convolution can be seen as another type of product and we derive a generalised convolution theorem. We conclude with a quick guide on implementing tensor contractions in python.

quant-ph↗

arXiv2018

Hamiltonians for one-way quantum repeaters

Quantum information degrades over distance due to the unavoidable imperfections of the transmission channels, with loss as the leading factor. This simple fact hinders quantum communication, as it relies on propagating quantum systems. A solution to this issue is to introduce quantum repeaters at regular intervals along a lossy channel, to revive the quantum signal. In this work we study unitary one-way quantum repeaters, which do not need to perform measurements and do not require quantum memories, and are therefore considerably simpler than other schemes. We introduce and analyze two methods to construct Hamiltonians that generate a repeater interaction that can beat the fundamental repeaterless key rate bound even in the presence of an additional coupling loss, with signals that contain only a handful of photons. The natural evolution of this work will be to approximate a repeater interaction by combining simple optical elements.

quant-ph↗

arXiv2016

Sub-fidelity as a measure of irreversibility of decoherence

Quantum correlations between two systems can be impaired by the presence of an environment, as it can make systems decohere. We wish to evaluate how much coherence has been irreversibly lost. To do so, we study two qubits which leak correlations into an arbitrarily large environment and we find that our ability to steer one of them to the most coherent set of states possible by acting on the other is given by the sub-fidelity of two conditional states of the environment. Sub-fidelity is a quantity that was already known, but that until now lacked an operational interpretation. Furthermore, we conjecture the existence of a family of measures, of which fidelity and sub-fidelity are the first two, which measure the irreversibility of decoherence as we gain access to more and more degrees of freedom of the enviroment.

quant-ph↗

arXiv2015

The Quantum Hilbert Hotel

In 1924 David Hilbert conceived a paradoxical tale involving a hotel with an infinite number of rooms to illustrate some aspects of the mathematical notion of "infinity". In continuous-variable quantum mechanics we routinely make use of infinite state spaces: here we show that such a theoretical apparatus can accommodate an analog of Hilbert's hotel paradox. We devise a protocol that, mimicking what happens to the guests of the hotel, maps the amplitudes of an infinite eigenbasis to twice their original quantum number in a coherent and deterministic manner, producing infinitely many unoccupied levels in the process. We demonstrate the feasibility of the protocol by experimentally realising it on the orbital angular momentum of a paraxial field. This new non-Gaussian operation may be exploited for example for enhancing the sensitivity of N00N states, for increasing the capacity of a channel or for multiplexing multiple channels into a single one.

quant-ph↗

arXiv2015

Recovering full coherence in a qubit by measuring half of its environment

When quantum systems interact with the environment they lose their quantum properties, such as coherence. Quantum erasure makes it possible to restore coherence in a system by measuring its environment, but accessing the whole of it may be prohibitive: realistically one might have to concentrate only on an accessible subspace and neglect the rest. If that is the case, how good is quantum erasure? In this work we compute the largest coherence $\langle \mathcal C\rangle$ that we can expect to recover in a qubit, as a function of the dimension of the accessible and of the inaccessible subspaces of its environment. We then imagine the following game: we are given a uniformly random pure state of $n+1$ qubits and we are asked to compute the largest coherence that we can retrieve on one of them by optimally measuring a certain number $0\leq a\leq n$ of the others. We find a surprising effect around the value $a\approx n/2$: the recoverable coherence sharply transitions between 0 and 1, indicating that in order to restore full coherence on a qubit we need access to only half of its physical environment (or in terms of degrees of freedom to just the square root of them). Moreover, we find that the recoverable coherence becomes a typical property of the whole ensemble as $n$ grows.

quant-ph↗

arXiv2014

Divergence of an orbital-angular-momentum-carrying beam upon propagation

There is recent interest in the use of light beams carrying orbital angular momentum (OAM) for creating multiple channels within free-space optical communication systems. One limiting issue is that, for a given beam size at the transmitter, the beam divergence angle increases with increasing OAM, thus requiring a larger aperture at the receiving optical system if the efficiency of detection is to be maintained. Confusion exists as to whether this divergence scales linarly with, or with the square root of, the beam's OAM. We clarify how both these scaling laws are valid, depending upon whether it is the radius of the Gaussian beam waist or the rms intensity which is kept constant while varying the OAM.

quant-ph↗

arXiv2014

The optimal bound of quantum erasure with limited means

In practical applications of quantum information science, quantum systems can have non-negligible interactions with the environment, and this generally degrades the power of quantum protocols as it introduces noise. Counteracting this by appropriately measuring the environment (and therefore projecting its state) would require access all the necessary degrees of freedom, which in practice can be far too hard to achieve. To better understand one's limitations, we calculate the upper bound of optimal quantum erasure (i.e. the highest recoverable visibility, or "coherence"), when erasure is realistically limited to an accessible subspace of the whole environment. In the particular case of a two-dimensional accessible environment, the bound is given by the sub-fidelity of two particular states of the \emph{inaccessible} environment, which opens a new window into understanding the connection between correlated systems. We also provide an analytical solution for a three-dimensional accessible environment. This result provides also an interesting operational interpretation of sub-fidelity. We end with a statistical analysis of the expected visibility of an optimally erased random state and we find that 1) if one picks a random pure state of 2 qubits, there is an optimal measurement that allows one to distill a 1-qubit state with almost 90\% visibility and 2) if one picks a random pure state of 2 qubits in an inaccessible environment, there is an optimal measurement that allows one to distill a 1-qubit state with almost twice its initial visibility.

quant-ph↗

arXiv2014

Limitations to the determination of a Laguerre-Gauss spectrum via projective, phase-flattening measurement

One of the most widely used techniques for measuring the orbital angular momentum components of a light beam is to flatten the spiral phase front of a mode, in order to couple it to a single-mode optical fiber. This method, however, suffers from an efficiency that depends on the orbital angular momentum of the initial mode and on the presence of higher order radial modes. The reason is that once the phase has been flattened, the field retains its ringed intensity pattern and is therefore a nontrivial superposition of purely radial modes, of which only the fundamental one couples to a single mode optical fiber. In this paper, we study the efficiency of this technique both theoretically and experimentally. We find that even for low values of the OAM, a large amount of light can fall outside the fundamental mode of the fiber, and we quantify the losses as functions of the waist of the coupling beam of the orbital angular momentum and radial indices. Our results can be used as a tool to remove the efficiency bias where fair-sampling loopholes are not a concern. However, we hope that our study will encourage the development of better detection methods of the orbital angular momentum content of a beam of light.

physics.optics↗

arXiv2014

Colloquium: Understanding Quantum Weak Values: Basics and Applications

Since its introduction 25 years ago, the quantum weak value has gradually transitioned from a theoretical curiosity to a practical laboratory tool. While its utility is apparent in the recent explosion of weak value experiments, its interpretation has historically been a subject of confusion. Here a pragmatic introduction to the weak value in terms of measurable quantities is presented, along with an explanation for how it can be determined in the laboratory. Further, its application to three distinct experimental techniques is reviewed. First, as a large interaction parameter it can amplify small signals above technical background noise. Second, as a measurable complex value it enables novel techniques for direct quantum state and geometric phase determination. Third, as a conditioned average of generalized observable eigenvalues it provides a measurable window into nonclassical features of quantum mechanics. In this selective review, a single experimental configuration to discuss and clarify each of these applications is used.

quant-ph↗

arXiv2014

A "fair sampling" perspective on an apparent violation of duality

In the event in which a quantum mechanical particle can pass from an initial state to a final state along two possible paths, the duality principle states that "the simultaneous observation of wave and particle behavior is prohibited". [M. O. Scully, B.-G. Englert, and H. Walther. Nature, 351:111-116, 1991.] emphasized the importance of additional degrees of freedom in the context of complementarity. In this paper, we show how the consequences of duality change when allowing for biased sampling, that is, postselected measurements on specific degrees of freedom of the environment of the two-path state. Our work contributes to the explanation of previous experimental apparent violations of duality [R. Menzel, D. Puhlmann, A. Heuer, and W. P. Schleich. Proc. Natl. Acad. Sci., 109(24):9314-9319, 2012.] and opens up the way for novel experimental tests of duality.

physics.optics↗