Indefinite causal order has attracted a growing attention during the last decade, with the quantum switch being its most prominent instance. The quantum switch has been implemented in several photonic experiments and has also provided computational advantages in several tasks. However, indefinite causal order on continuous-variable (CV) quantum systems has been barely studied. This thesis suggests that applications of indefinite causal order on CV quantum systems may offer broader advantages than using finite-dimensional quantum systems. We follow an exploratory approach comparing the performance of the quantum switch with high-dimensional and CV quantum systems in some applications to quantum computing and quantum metrology. In particular, we use Complex Hadamard matrices to introduce a generalised family of promise problems, which reduces to the known Fourier and Hadamard promise problems as limiting cases. We show that the quantum switch provides query advantage for both the continuous and discrete cases, and prove that a CV system is necessary for implementing the most general promise problem. In addition, we propose an estimation procedure for d-dimensional unitary transformations using the quantum switch. For d > 2, the unitary transformations close to the identity are estimated saturating the quantum Cram´ er-Rao bound. For d = 2, the estimation of all unitary transformations is also optimal with some prior information. We replace the unitary transformation by an arbitrary quantum channel and show that our procedure encodes the χ matrix of the channel in the output state of the control system. Although indefinite causal order does not offer metrological advantage in our proposal, it is worthy to note that an extension to the CV regime would require an infinite-dimensional control system. This inspires the introduction of the quantum switch with continuous control, which leads to new phenomena such as the accelerando/rallentando effect. We conclude that using CV systems enlarges the number of tasks benefited by indefinite causal order and exhibits new phenomena within the field.
Indefinite causal order has attracted a growing attention during the last decade, with the quantum switch being its most prominent instance. The quantum switch has been implemented in several photonic experiments and has also provided computational advantages in several tasks. However, indefinite causal order on continuous-variable (CV) quantum systems has been barely studied. This thesis suggests that applications of indefinite causal order on CV quantum systems may offer broader advantages than using finite-dimensional quantum systems. We follow an exploratory approach comparing the performance of the quantum switch with high-dimensional and CV quantum systems in some applications to quantum computing and quantum metrology. In particular, we use Complex Hadamard matrices to introduce a generalised family of promise problems, which reduces to the known Fourier and Hadamard promise problems as limiting cases. We show that the quantum switch provides query advantage for both the continuous and discrete cases, and prove that a CV system is necessary for implementing the most general promise problem. In addition, we propose an estimation procedure for d-dimensional unitary transformations using the quantum switch. For d > 2, the unitary transformations close to the identity are estimated saturating the quantum Cram´ er-Rao bound. For d = 2, the estimation of all unitary transformations is also optimal with some prior information. We replace the unitary transformation by an arbitrary quantum channel and show that our procedure encodes the χ matrix of the channel in the output state of the control system. Although indefinite causal order does not offer metrological advantage in our proposal, it is worthy to note that an extension to the CV regime would require an infinite-dimensional control system. This inspires the introduction of the quantum switch with continuous control, which leads to new phenomena such as the accelerando/rallentando effect. We conclude that using CV systems enlarges the number of tasks benefited by indefinite causal order and exhibits new phenomena within the field.
Research on indefinite causal structures is a rapidly evolving field that has a potential not only to make a radical revision of the classical understanding of space-time but also to achieve enhanced functionalities of quantum information processing. For example, it is known that indefinite causal structures provide exponential advantage in communication complexity when compared to causal protocols. In quantum computation, such structures can decide whether two unitary gates commute or anticommute with a single call to each gate, which is impossible with conventional (causal) quantum algorithms. A generalization of this effect to n unitary gates, originally introduced in Ara\'ujo et al. [Phys. Rev. Lett. 113, 250402 (2014)] and often called Fourier promise problem (FPP), can be solved with the quantum-n-switch and a single call to each gate, while the best known causal algorithm so far calls O(n2) gates. In this article, we show that this advantage is smaller than expected. In fact, we present a causal algorithm that solves the only known specific FPP with O(nlog(n)) queries and a causal algorithm that solves every FPP with O(nn) queries. Besides the interest in such algorithms on their own, our results limit the expected advantage of indefinite causal structures for these problems.
The quantum switch, the canonical example of a process with indefinite causal order, has been claimed to provide various advantages over processes with definite causal orders for some particular tasks in the field of quantum metrology. In this work, we argue that some of these advantages in fact do not hold if a fairer comparison is made. To this end, we consider a framework that allows for a proper comparison between the performance, quantified by the quantum Fisher information, of different classes of indefinite causal order processes and that of causal strategies on a given metrological task. More generally, by considering the recently proposed classes of circuits with classical or quantum control of the causal order, we come up with different examples where processes with indefinite causal order offer (or not) an advantage over processes with definite causal order, qualifying the interest of indefinite causal order regarding quantum metrology. As it turns out, for a range of examples, the class of quantum circuits with quantum control of causal order, which are known to be physically realizable, is shown to provide a strict advantage over causally ordered quantum circuits as well as over the class of quantum circuits with causal superposition. Thus, the consideration of this class provides new evidence that indefinite causal order strategies can strictly outperform definite causal order strategies in quantum metrology.
Higher-order maps account for the most general transformations in quantum mechanics—they transform input quantum maps to output quantum maps. In this thesis, I focus on the application of such higher-order maps representing two important processes: (1) a particular type of non-Markovian quantum dynamics, and (2) indefinite causal order. First, I will introduce our work on the non-Markovian evolution of quantum states. Such a situation occurs when the past system-environment interaction leads the environment to memorize the information about the system. The measure of the non-Markovianity is given by the temporal correlation among the operations on the system. One way to estimate such correlation is to employ a full tomography of the process matrix, which is computationally intractable. Instead, we utilize machine learning algorithms without using tomographically complete measurement, to estimate the measure of non-Markovianity. We apply our model to a dataset obtained from an optical experiment and show that we are able to predict the measure of non-Markovianity with 90% accuracy. I then introduce various aspects of indefinite causal order. Ordinary experience points only to a definite causal structure, but quantum mechanics admits a more general causal structure that allows for the superposition of different causal orders. That means that the statements, is in the causal past of event B, and B is in the causal past of event A can both be true. The quantum switch can physically realise such an indefinite causal structure where a control quantum system controls the order of operations acting on a target quantum system. Superposition in the control leads to the superposition of the order. We experimentally realise a quantum switch with the control qubit being the polarisation of light and the target qubit being the transverse spatial mode of light. We first verify our quantum switch by measuring a `causal witness'. We then demonstrate an advantage of the quantum switch for classical communication. Finally, I present our work on classical communication through the most general quantum causal structures, represented by the process matrix. We formulate different classical capacities of a bi-partite process matrix. We also study one-way and bidirectional communication protocols through such a process and establish the relevant communication bounds. These enable further analyses of higher-order maps that take advantage of the full suite of processes that quantum mechanics allows.
Processes with indefinite causal orders (ICOs), such as the quantum switch, are higher-order quantum processes that superpose the order in which quantum operations are performed. Such coherent control yields computational advantages but is not faithfully captured by existing quantum programming languages: either they are restricted to the unitary case, and thus cannot combine ICOs with measurement, or they treat coherent control nonlinearly. In both cases, they do not realize the full computational power of ICOs. We introduce a higher-order quantum functional language that supports general quantum computation, not merely the permutation of channels, and whose linear type system allows quantum control to be well-defined beyond the unitary case, on arbitrary quantum channels. We equip this language with a small-step operational semantics that synchronizes measurement outcomes across superposed branches, using device references and a memory function. We also give a denotational semantics by means of completely positive maps. With linearity as the only constraint, some well-typed terms would denote unphysical maps. We therefore impose a typing discipline that goes beyond linearity, and interpret programs in the causal category Caus[CPM], under which every well-typed program is physically meaningful, a property that can be checked statically and efficiently. We prove soundness, and study the language's expressive power: it can express every quantum channel at first order, and at second order a large subclass of the so-called quantum circuits with quantum control (QC-QCs), containing the quantum switch. Last but not least, we show that this language is well-designed enough to be extended to the nonlinear setting with recursion.
Quantum compositions of channels can grant advantages over classical compositions in quantum information processing or thermodynamical tasks. One of such quantum composition is one with indefinite causal order, where multiple quantum channels are placed in a superposition of different causal or operation orders. The advantages of indefinite causal order were demonstrated in an experimentally realizable setup known as the quantum switch and were often attributed to this new quantum resource of indefinite causality, but some doubts and disagreements still remains as compositions without indefinite causal order can achieve the same advantages. In this work, we proposed that the quantum switch's uniqueness lies in its ability to exhibit coherent non-Markovianity, where memory effects acts across different paths of superposition. We showed that the quantum switch violates completely positive or CP-divisibility and thus has intrinsic non-Markovianity in its evolution, which allows for non-Markovian backflow of information from the environment. This backflow of information can then play a role in the many advantages seen in the quantum switch, and by extending the quantum switch operation such that the non-Markovianity is controllable, we showed that these advantages depend on the presence and amount of non-Markovianity, achieving the quantum switch case when the system is fully non-Markovian. Particularly, we demonstrated this for the quantum switch's advantages in communication capacities, work extraction via daemonic ergotropy, and refrigeration via heat extraction. Other quantum compositions that are often compared to indefinite causal order, such as the superposition of independent channels and superposition of trajectories, are also captured by this extended construction of the quantum switch, and we showed that their difference with the quantum switch lies in the presence or absence of non-Markovianity. Our work suggests that non-Markovianity is an important recipe for the quantum switch or indefinite causal order, and is a nonnegligible factor for its advantages.
The quantum switch is a quantum computational primitive that provides computational advantage by applying operations in a superposition of orders. In particular, it can reduce the number of gate queries required for solving promise problems where the goal is to discriminate between a set of properties of a given set of unitary gates. In this work, we use Complex Hadamard matrices to introduce more general promise problems, which reduce to the known Fourier and Hadamard promise problems as limiting cases. Our generalization loosens the restrictions on the size of the matrices, number of gates and dimension of the quantum systems, providing more parameters to explore. In addition, it leads to the conclusion that a continuous variable system is necessary to implement the most general promise problem. In the finite dimensional case, the family of matrices is restricted to the so-called Butson-Hadamard type, and the complexity of the matrix enters as a constraint. We introduce the “query per gate'' parameter and use it to prove that the quantum switch provides computational advantage for both the continuous and discrete cases. Our results should inspire implementations of promise problems using the quantum switch where parameters and therefore experimental setups can be chosen much more freely.