Classical simulation of quantum circuits

Project description

Started January 2020

Background: For the future development of quantum technologies, it is crucial for us to understand what components of the quantum theory are responsible for quantum supremacy, i.e. the potential ability of quantum computers to solve problems that cannot be solved efficiently on classical machines. One of the most promising ways to achieve this is to identify sub-theories of the quantum theory that can be efficiently simulated on classical computers. The first result of this kind was the celebrated Gottesman-Knill theorem, which states that the stabiliser sub-theory, where one is restricted to state preparation and measurements in the computational basis and evolution according to Clifford gates, can be simulated in such a way. Moreover, the addition of a single type of a pure “magic” (non-stabiliser) state allows one to promote this classically simulable sub-theory to universal quantum computing, making magic states a proper resource for quantum computation when Clifford gates are considered free (easy to implement experimentally). 

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Goals: We want to develop a unified scheme for classical simulation of universal quantum circuits based on a three-step algorithm: identifying a free sub-theory, gadgetizing all resources (i.e. replacing non-free quantum gates with resource states), sampling and propagating the free states taken from optimal decomposition of resource states into free states of the theory. This three-step algorithm should unify many known simulation schemes, thus deepening our understanding of the nature of quantum computing, but also will provide a clear way to develop novel simulation algorithms. First, we want to apply this approach to build a fast classical algorithm to simulate Clifford+T circuits. We then want to extend it to circuits built from Gaussian+non-Gaussian gates and matchgate circuits with a resource SWAP gate. Finally, we would like to design a fully universal treatment of the problem. Complementary to this, we aim at implementing our algorithms on classical computers and use the developed software to verify near-term intermediate scale quantum devices.

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Resource theories and quantum communication

Project description

Started May 2019

​Background: Communication problems lie at the very heart of quantum information science, with protocols such as quantum teleportation and super-dense coding capturing the essence of quantum information processing. A typical communication scenario consists of encoding a message in a quantum system, sending it via a channel, and then decoding it on the other side. All three stages require a fine control over quantum systems and the ability to manipulate them efficiently. It is then very natural to ask: how would the communication be affected, if the control is not perfect or the state manipulation is constrained? 

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Goals: Our aim will be to describe constrained communication scenarios in the language of resource theories, and then apply its formalism to relate constrained communication rates to appropriate quantum resources. On a more technical side, we would like to develop physically relevant resource theories of channels, which would provide useful quantifiers for important information processing tasks such as error correction or entanglement sharing.

Dissipation of quantum resources

Project description

Started March 2020

​Background: In principle, while processing quantum information, any initial state can be transformed into any final state. One could thus conclude that all quantum states are equally valuable or resourceful. In reality, however, some transformations are harder to implement than others, which results in a partial ordering of the set of quantum states, with the hardest to prepare at the top, and easiest at the bottom. Such a resource hierarchy arises naturally when we face any kind of restriction: from the locality constraint, through experimental difficulties in preparing particular superpositions, to fundamental constraints induced by physical laws like energy conservation. Moreover, these constraints on processing quantum information result in irreversibility, i.e. during the interconversion process some resource content is unavoidably lost. From both the fundamental and applied perspective it is then important to understand the nature and limits of the dissipation of quantum resources.

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Goals: We would like to develop a general framework allowing one to quantitatively study resource dissipation for various resource theories (thermodynamics, entanglement, magic, etc.), and to characterise optimal state transformation protocols that minimise dissipation. We would also like to explore the phenomenon of resource resonance (leading to lossless interconversion) and design experimental setups employing it. Finally, we aim at employing our results to derive general fluctuation-dissipation relations for quantum resources, thus generalising the known thermodynamic phenomenon to a general resource-theoretic setting.

Quantum coherence and memory effects

Project description

Started June 2019

Background: Although all fundamental interactions are memoryless, the basic information processing primitives (such as the bit-flip operation) cannot be performed classically in a time-continuous fashion without employing a memory. However, this picture changes dramatically if instead we consider memoryless quantum dynamics. This is due to quantum coherence, arising from the superposition principle, which can effectively act as an internal memory of the system during the evolution. 

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Goals: We want to explore potential quantum advantages arising from coherence acting as a memory, both in thermodynamic and information-processing scenarios (e.g. enhanced cooling or exponential improvement in space-time cost of realising a given process). On a more mathematical side, we would like to characterise quantum embeddable stochastic processes, i.e. these processes that can be implemented quantumly without employing a memory. Finally, we will also try to employ the developed framework to bridge the gap between the resource theories and control theory.

Structural differences between classical and quantum randomness

Project description

Started February 2020

Background: Random processes are ubiquitous in both classical and quantum physics. However, the nature of randomness in these two regimes differs significantly. On the one hand, classical random evolution is necessarily irreversible. On the other hand, quantum evolution may be completely deterministic (and thus reversible if no measurement is performed), but nevertheless lead to random measurement outcomes of observable A by transforming a system into a coherent superposition of eigenstates of A. When probing the dynamics of the system one can therefore observe the same random transitions, irrespectively of whether the evolution is coherent or incoherent. The questions then arise: to what extent an observed random transformation can be explained via the underlying
deterministic and coherent process, and how much unavoidable classical randomness must be involved in it?

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Goals: Our main goal will be to study what kind of transition matrices can be induced by reversible unitary dynamics, i.e. we wish to understand the structure of unistochastic matrices. These technical results will not only characterise random processes with a potentially deterministic cause, but could also be employed to the studies of quantum walks on graphs