Addressable quantum gates

TL;DR

The paper proposes Addressable Quantum Circuits (AQC), a generalization of textbook quantum circuits where wiring is encoded inside gates via address registers, enabling superpositions of causal orders and explicit circuit geometry. It demonstrates that AQCs can realize indefinite causal orders, exemplified by encoding the quantum switch and a polarizing beam splitter within the model. A central result is the full characterization of nameblind operations—gate operators that commute with address renamings—showing that such gates decompose into blocks constructed from nameblind matrices, preserving locality while allowing nontrivial dynamics. The work situates AQCs within the landscape of quantum causal models (e.g., QCGD) and outlines compositional grammars for parallel, sequential, and general connectivity, offering a concrete, constructive framework for distributed quantum computation with quantum-evolving connectivity. Overall, the approach clarifies how renaming invariance constrains processing and provides a rigorous path toward implementing and reasoning about dynamic quantum wirings.

Abstract

We extend the circuit model of quantum computation so that the wiring between gates is soft-coded within registers inside the gates. The addresses in these registers can be manipulated and put into superpositions. This aims at capturing indefinite causal orders and making their geometrical layout explicit: we express the quantum switch and the polarizing beam-splitter within the model. In this context, our main contribution is a full characterization of the anonymity constraints. Indeed, the names used as addresses should not matter beyond the wiring they describe, i.e. quantum evolutions should commute with "renamings". We show that these quantum evolutions can still act non-trivially upon the names. We specify the structure of "nameblind" matrices.

Figures (16)

• Figure 1: Dashed boxes represent sectors. (a) Textbook quantum circuits require a background layer that implies a time ordering, here shown explicitly in light purple. (b) The AQC formalism frees the information flow from the background surface. Each sector has a target space, holding a target address, as represented by the arrows. Addresses could have been taken in any other integer set instead.
• Figure 2: (a) A sector. (b) Stored addresses vs. quantum data. (c) Input vs. output spaces. When all output (resp. input) spaces are empty, we only represent the input (resp. output) spaces, as in Fig. \ref{['fig:QSAQC']}.
• Figure 3: Effect of the transport step upon two connected sectors ---see example \ref{['example:transport']}. Here and elsewhere in the paper arrows represent the content of each target space. If the state of the target space is a superposition of multiple addresses, one arrow points to each target.
• Figure 4: Superposing causal orders by superposing addresses.
• Figure 5: Each step corresponds to one application of the global evolution operator $G=TS$. The step $n.5$ for any integer $n$ corresponds to applying the operator $S$ to the state of step $n$. The control qubit is $\ket c = \alpha\ket 0+\beta\ket 1$. Blue represents a scalar coefficient of $\alpha$ before the basis state, red represents a coefficient of $\beta$ and black represents $1$. When nothing is specified in caption, only input spaces are represented, whereas "Output spaces" in caption indicates that only the output spaces are represented. A ${{\mathcal{O}}}$ in subscript (as in Figs. \ref{['fig:qsaqc:1']}, \ref{['fig:qsaqc:2']}, and \ref{['fig:qsaqc:9']}) indicates that a specific element is in the output space.

Theorems & Definitions (45)

• definition 1: Target address space, stored address space
• definition 2: Data space
• definition 3: Input and output space, sector space
• definition 4: Gates, Gate Operators
• definition 5: Skeleton, Addressable Quantum Circuit
• definition 6: Evolution
• definition 7: Renaming and nameblind NAQC
• theorem 1: Nameblind AQC
• theorem 2: $(n,n)$-nameblind matrices
• corollary 1: Pure nameblind matrices