Causal and compositional structure of unitary transformations

Robin Lorenz1,2 and Jonathan Barrett1

1Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK
2Cambridge Quantum Computing Ltd, 17 Beaumont Street, Oxford OX1 2NA, UK

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The causal structure of a unitary transformation is the set of relations of possible influence between any input subsystem and any output subsystem. We study whether such causal structure can be understood in terms of compositional structure of the unitary. Given a quantum circuit with no path from input system $A$ to output system $B$, system $A$ cannot influence system $B$. Conversely, given a unitary $U$ with a no-influence relation from input $A$ to output $B$, it follows from [B. Schumacher and M. D. Westmoreland, Quantum Information Processing 4 no. 1, (Feb, 2005)] that there exists a circuit decomposition of $U$ with no path from $A$ to $B$. However, as we argue, there are unitaries for which there does not exist a circuit decomposition that makes all causal constraints evident $\textit{simultaneously}$. To address this, we introduce a new formalism of `extended circuit diagrams', which goes beyond what is expressible with quantum circuits, with the core new feature being the ability to represent direct sum structures in addition to sequential and tensor product composition. A $\textit{causally faithful}$ extended circuit decomposition, representing a unitary $U$, is then one for which there is a path from an input $A$ to an output $B$ if and only if there actually is influence from $A$ to $B$ in $U$. We derive causally faithful extended circuit decompositions for a large class of unitaries, where in each case, the decomposition is implied by the unitary's respective causal structure. We hypothesize that every finite-dimensional unitary transformation has a causally faithful extended circuit decomposition.

Whenever one is able to draw an intuitive picture that succinctly captures the essential features of a complicated whole, then not only does it usually help communication, arguably, this also is a sign of having achieved good conceptual understanding. Also for quantum theory there is a rich history of developing and employing diagrammatic reasoning, where quantum circuit diagrams capture aspects of how quantum systems interact with each other and evolve over time. Circuit diagrams are the basic ingredient to one of the main paradigms of quantum computation, and have also proven useful in research ranging from the foundations of quantum theory to the design of algorithms and efficient quantum compilers. One particular area of research of foundational, as well as applied importance is causality. Unitary transformations that play a fundamental role in the quantum formalism, describe the evolution of a set of systems and have a clear notion of causal structure: which of the systems can causally influence which other systems.

Combining all these lines of thought, this paper asks: can one understand the causal structure of a unitary transformation in compositional terms, i.e. through an intuitive diagram, where its components are connected up in such a way that lays bare where and how causal influence goes? We first show that this is generally not possible with ordinary circuit diagrams. We then derive new kinds of decompositions for many causal structures, as well as introduce a new graphical language of 'extended circuit diagrams' to visualise them. This novel perspective to study causal structure and its ramifications is of conceptual significance, as well as likely to facilitate progress with other open problems, which it in fact already has in the study of indefinite causal order, a much debated area of quantum physics.

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Cited by

[1] Paolo Perinotti, "Causal influence in operational probabilistic theories", Quantum 5, 515 (2021).

[2] Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov, "Quantum Causal Models", arXiv:1906.10726.

[3] Wataru Yokojima, Marco Túlio Quintino, Akihito Soeda, and Mio Murao, "Consequences of preserving reversibility in quantum superchannels", arXiv:2003.05682.

[4] Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov, "Cyclic quantum causal models", Nature Communications 12, 885 (2021).

[5] Matt Wilson and Augustin Vanrietvelde, "Composable constraints", arXiv:2112.06818.

[6] Augustin Vanrietvelde, Hlér Kristjánsson, and Jonathan Barrett, "Routed quantum circuits", arXiv:2011.08120.

[7] Matt Wilson, Giulio Chiribella, and Aleks Kissinger, "Quantum Supermaps are Characterized by Locality", arXiv:2205.09844.

[8] Nick Ormrod, Augustin Vanrietvelde, and Jonathan Barrett, "Causal structure in the presence of sectorial constraints, with application to the quantum switch", arXiv:2204.10273.

[9] Augustin Vanrietvelde, Nick Ormrod, Hlér Kristjánsson, and Jonathan Barrett, "Consistent circuits for indefinite causal order", arXiv:2206.10042.

[10] Pablo Arrighi, Christopher Cedzich, Marin Costes, Ulysse Rémond, and Benoît Valiron, "Addressable quantum gates", arXiv:2109.08050.

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