Causal Decompositions of 1D Quantum Cellular Automata

Augustin Vanrietvelde; Octave Mestoudjian; Pablo Arrighi

Causal Decompositions of 1D Quantum Cellular Automata

T0 review · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A one-dimensional quantum cellular automaton of causality radius $r$ is exactly a layered circuit of nearest-neighbour unitary interactions on lines longer than $4r$.

desk verdict First constructive form for 1D QCAs of arbitrary radius (N>4r), proved as an iff with routed nearest-neighbour circuits; looks right, but the circuit-form bridge leans on a companion paper. read the letter →

arxiv 2506.22219 v1 pith:U7LKKBQ5 submitted 2025-06-27 quant-ph math-phmath.MP

Augustin Vanrietvelde , Octave Mestoudjian , Pablo Arrighi This is my paper

classification quant-phmath-phmath.MP MSC 81P6846L05

keywords 1Dquantumcellularautomatacausaldecompositionsroutedunitarycircuitsnon-factorC*-algebrasvertexsplittingtranslation-invariantQCAconstructiveform

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

On a finite line of $N$ quantum systems with $N>4r$, the paper proves that causality and composability coincide: a unitary channel is a 1D quantum cellular automaton of radius $r$ if and only if it can be written as a routed unitary circuit of nearest-neighbour interactions. This is the first constructive form for QCAs of radius one or more, not just radius $\tfrac{1}{2}$, and it comes with no ancillas, no extra radius, and no pairwise-commutation promises. The same statement holds in translation-invariant form when the original QCA is translation invariant. A sympathetic reader should care because it turns the abstract causal constraint defining a QCA into an explicit recipe for building exactly those unitaries, showing that in finite 1D systems causal structure and compositional structure are the same thing.

What carries the argument

The load-bearing object is a partition of a finite-dimensional C*-algebra into sub-C*-algebras indexed by subsets of a graph, where parts need not be tensor factors; non-factorness is tracked by the atomic projectors of centres. The proof's engine is the vertex-splitting lemma (Lemma D.9), which says that under a correlation-length bound a single site of a strongly connected 1D partition can be replaced by two sites, and this operation iterated gives a fine-graining theorem: a radius-$r$ QCA can be interleaved with fine and coarse grainings to reduce its radius by $\tfrac{1}{2}$ at each step. To move from the algebraic picture to unitaries, the paper uses index-matching routed circuits: wires carry sector labels and boxes are routed unitaries, and a consistency condition (each repeated index has one start and one end) guarantees the whole diagram is a single unitary. The key bridge is full representability, meaning every partition arising in the construction can be concretely realised as algebras of routed maps over sectorised Hilbert spaces, together with a null-sector trick allowing zero-dimensional sectors.

What would settle it

One concrete check is to search the finite set of radius-1 unitaries on $N=5$ sites and compare them against all Figure 1 circuits: if any unitary passes the causality condition but provably fails the algebra conditions of Lemma D.9 at the first vertex split, the 'only if' direction of Theorem 4.2 collapses.

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Extended reading notes

Core claim

The paper's central claim is Theorem 4.2: for $N>4r$, a unitary map between tensor-product Hilbert spaces over a line or loop is a 1D QCA of radius $r$ if and only if it decomposes as the routed unitary circuit of Figure 1 — a stack of $2r$ layers of nearest-neighbour routed unitaries, each layer a fine-graining followed by a coarse-graining. Algebraically (Theorem 3.1) this is the statement that every such QCA factors as $U = U_{2r}\cdots U_1$ with each $U_i$ a radius-$\tfrac{1}{2}$ QCA, and each $U_i = U_i^{\mathrm{coarse}} U_i^{\mathrm{fine}}$. Theorem 4.3 adds that a translation-invariant QCA admits the same circuit with all boxes in a layer identical. The results give the first fully constructive form for 1D QCAs of arbitrary radius, and they show that the decomposition is exact: no ancillary wires, no inflated radius, no extra commutation assumptions are needed.

Load-bearing premise

The load-bearing premise is that the abstract partition data produced by the construction can always be drawn as explicit wire-and-box circuit data; this representation theorem is imported from a companion paper rather than proved here, and if it fails for any of the partitions the main circuit conclusion gives way.

Editorial extensions

If this is right

Any 1D QCA of radius $r$ on $N>4r$ sites can be implemented exactly by a layered routed circuit of nearest-neighbour unitaries, with no ancillas and no commuting sub-circuit promises.
Listing QCAs of radius $r$ becomes the same problem as classifying the Figure 1 routed circuits, and the translation-invariant case is classified by one routed unitary per layer.
The radius-$\tfrac{1}{2}$ form is no longer a special case: it becomes the atomic step of a universal decomposition for all radii.
A constructive implementation of a QCA no longer has to use coarse-graining tricks that silently increase the causality radius to $2r-\tfrac{1}{2}$; the layered form respects the original radius.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The $N>4r$ bound is probably not sacred: the proof's correlation-length estimates suggest the true threshold may be closer to $2r+2$, where the causality condition becomes non-trivial, and the narrow-window case $2r+2 \le N \le 4r$ is left open.
If the finite-line construction extends to infinite lattices through a uniform version of the same argument, the route would also give a constructive normal form for translation-invariant matrix-product unitaries in 1D.
The algebraic methods used here — partitions and centre-tracking — are natural tools for symmetry-constrained or fermionic QCAs, whose decompositions may be obtained by adapting the same vertex-splitting argument.
A testable extension is to use the decomposition to search for new 1D QCA invariants: any quantity invariant under the local routed boxes becomes an invariant of the whole QCA.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know up front. First, this paper delivers what it claims: for N > 4r, a finite 1D QCA of radius r is a QCA iff it decomposes into a 2r-layer routed circuit of nearest-neighbour unitaries, and the translation-invariant version holds with all boxes in a layer identical. That is the first constructive form beyond radius 1/2, it preserves the radius exactly (the index-theory route only gives 2r−1/2), and it comes with no ancillas and no commuting-promise caveats. Second, the central algebraic work — Theorem 3.1, the decomposition into radius-1/2 fine- and coarse-graining maps — is proved in this text, and the proof chain in Appendices D and E is careful and long. But the last step, turning that algebraic decomposition into the concrete routed circuit of Figure 1, imports representation theorems (full representability, Theorems F.1–F.3) from the authors' concurrent companion paper [21], and those are not proved here. The null-sector trick that extends the routed-map formalism to zero-dimensional sectors gets one line of justification. If the companion theory has a gap, the circuit-form theorem stated in the abstract would not be established, even though the algebraic decomposition would survive. That is the one load-bearing soft spot, and I would call it structural rather than a demonstrated flaw. The paper is upfront about the dependency, the stress-test found no circularity in the main equivalence, and the stated bounds and open cases (narrow 1D QCAs with 2r+2 ≤ N ≤ 4r, infinite lattices, fermionic, 2D) are handled honestly. The other soft spots are minor. Who is this for: anyone working on QCA structure, index theory, or causal decompositions. It is a genuine theorem with a plausible proof; given the Schumacher–Werner history in higher dimensions, independent checking of the long lemma chain is warranted before using the result as a black box. I would send it to a serious referee rather than desk reject, with instructions to verify the companion-paper dependency and the null-sector extension, and preferably to get a second opinion. My own inclination is conditional accept: make the needed representation theorems self-contained or ensure [21] is publicly verified, then the result should stand.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the causal-decomposition theorem is not assumed in the proof; reliance on the companion paper is external, parameter-free support rather than a self-referential reduction.

full rationale

The derivation chain is not circular. Theorem 3.1, the algebraic decomposition, is proved in the paper itself via vertex-splitting (Lemma D.9), fine-graining (Theorem D.1), and the induction in Section D.4; the decomposition is constructive and is not an input to its own proof. The conversion to routed circuits in Theorem 4.2 uses Theorem F.4, whose 'only if' direction invokes representability of the intermediate partitions (Theorems F.1 and F.3). That representability is imported from the authors' companion paper [21] (Theorems 5.1, 5.2, Proposition 4.3), and the 'null-sector trick' (Proposition F.1) is asserted rather than re-derived. These are legitimate provenance and verification concerns, but not circularity: [21] is a parameter-free general theory of partitions whose stated assumptions do not include the present causal-decomposition result, so by the standard for independent support it does not raise the circularity score. The 'if' direction of Theorem 4.2 is handled by the external soundness result Proposition 4.2 from Refs. [5,6]. No quantity is fitted to data and later relabeled as a prediction, no equation defining the target result is fed back into the argument, and no uniqueness theorem is invoked to forbid alternatives. Footnote 23's observation that many automorphisms in the proof are identities is a proof-strategy remark, not an admission that the theorem is assumed. The conclusion's explicit exclusion of narrow QCAs (2r+2 ≤ N ≤ 4r) delimits the scope of the claim and further confirms that the theorem is not vacuously or definitionally true. The only caveat worth flagging is the unverified meta-claim in Proposition F.1 that the proofs of [21] never use non-nullness; if that proved false, the bridge from the algebraic decomposition to the concrete routed-circuit form would need repair, but that is a correctness/provenance risk, not a circular step.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

No fitted constants appear anywhere: the theorems are universal over finite-dimensional Hilbert spaces, with N and r as inputs, and correlation length is a structural quantity, not a fitted parameter. The proof relies on standard finite-dimensional C*-algebra facts, on external results for the causal and compositional equivalence of two-part channels [4] and for circuit soundness [5,6], and on the authors' own companion-paper representation theory [21], which is the only self-sourced, not-yet-published load-bearing input. The invention ledger contains one entry: the partition framework itself, which is mathematical rather than physical.

assumptions (4)

standard math Finite-dimensional C*-algebra structure theory: centres decompose into atomic projectors, each block is a factor, and the von Neumann bicommutant theorem holds (Theorems 2.1-2.2, Proposition B.2).
Used throughout to decompose algebras into blocks and to identify double commutants; standard textbook material [41].
domain assumption The equivalence between absence of causal influence and the compositional (causal decomposition) form for two-input two-output unitary channels (Ref [4]).
Anchors the paper's definition and motivation of causal decompositions and the program of Ref [5]; treated as an established external result.
domain assumption Consistent index-matching circuits of routed unitaries are unitary, and path absence in such circuits implies absence of causal influence (Refs [6,5], Propositions 4.1 and 4.2).
Used for the 'if' direction of Theorem 4.2 and for the soundness of the circuit formalism; imported from the authors' own prior routed-circuit papers and from Ref [5].
ad hoc to paper The representation theory of partitions from the companion paper: every partition admits a concrete represented form (Ref [21] Theorem 5.2), routes are linked to atomic projectors of the centres of interval algebras (Ref [21] Theorems 5.1 and Proposition 4.3), and all partitions arising in Theorem…
Load-bearing for the bridge from the algebraic decomposition (Theorem 3.1) to the unitary routed-circuit form (Theorem 4.2). The present paper does not reproduce these proofs, only states and uses them.

invented entities (1)

Partition framework: a family of sub-C* algebras (A_S)_{S subset of X} of a possibly non-factor finite-dimensional C* algebra, indexed by all subsets, with bipartition conditions (Definition 2.5). independent evidence
purpose: Provides the language in which fine-grainings, coarse-grainings and the causal decomposition are defined and proved (Sections 2-3, Appendices D and F).
The framework is developed in the authors' concurrently submitted companion paper [21] and used as a tool here; its correctness is testable through the paper's own theorems (for instance the radius-1/2 case reproduces known results) and through explicit decompositions for small N. It is a mathematical entity, not a physical one, so the 'graviton problem' does not apply directly; it is flagged because the framework is new and self-sourced.

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Pith. "Pith review of Causal Decompositions of 1D Quantum Cellular Automata." pith.science (2026). https://pith.science/paper/U7LKKBQ5

@misc{pith2026250622219,
  author       = {Pith},
  title        = {Pith review of: Causal Decompositions of 1D Quantum Cellular Automata},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U7LKKBQ5}},
  note         = {Machine review of arXiv:2506.22219}
}

abstract

Understanding quantum theory's causal structure stands out as a major matter, since it radically departs from classical notions of causality. We present advances in the research program of causal decompositions, which investigates the existence of an equivalence between the causal and the compositional structures of unitary channels. Our results concern one-dimensional Quantum Cellular Automata (1D QCAs), i.e. unitary channels over a line of $N$ quantum systems (with or without periodic boundary conditions) that feature a causality radius $r$: a given input cannot causally influence outputs at a distance more than $r$. We prove that, for $N \geq 4r + 1$, 1D QCAs all admit causal decompositions: a unitary channel is a 1D QCA if and only if it can be decomposed into a unitary routed circuit of nearest-neighbour interactions, in which its causal structure is compositionally obvious. This provides the first constructive form of 1D QCAs with causality radius one or more, fully elucidating their structure. In addition, we show that this decomposition can be taken to be translation-invariant for the case of translation-invariant QCAs. Our proof of these results makes use of innovative algebraic techniques, leveraging a new framework for capturing partitions into non-factor sub-C* algebras.

Figures

Figures reproduced from arXiv: 2506.22219 by the authors.

**Figure 2.** ΓN , Γ +1/2 N , and Γ fine N . and l > 0, we denote S ± l := {m ∈ Γ | ∃n ∈ S, d(m, n) ≤ l}. Definition 2.6 (Generalised 1D quantum cellular automaton). Let N ∈ N and r a positive integer or half-integer. We denote Γ out := ΓN when r is integer and Γ out := Γ+1/2 N when it is half-integer. Let (AS)S⊆ΓN ⊢ ω in , (BS)S⊆Γout ⊢ ω out be partitions of two (finite-dimensional) factor C* algebras. A quantum cellular automa… view at source ↗

**Figure 3.** Vertex-splitting. Lemma D.9 indicates what is a sufficient data specification to turn a partition over ΓN to one over Γ split, in which the 0 vertex has been split into two vertices ± 1 4 . The distances d(0, q9), d(q9, p9), d(0, q+), d(q+, p+) are all greater or equal to the correlation length l. AJ0,p+K ∩ A′ Jq9,− 1 4 K = AJ0,p+K ∩ AJq9,p+K ∩ A′ Jq9,− 1 4 K = AJ0,p+K ∩ (Zq9−1 ∩ Zq9 ) ∨ AJ 1 4 ,p+K = AJ0,p+K ∩ … view at source ↗

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Reference graph

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D. R. Farenic, Algebras of Linear Transformations. Springer New York, 2001. A Notations In this Appendix, we recap the notations used through the paper. All algebras are finite- dimensional, and all sets of indices are finite. Algebras C*-algebras are denoted using curly lette...

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= 1 2, even though 1 4 and 3 4 are ‘one node away’ from each other. 12With a slight abuse of notation, we sometimes use this notation across different graphs: for instance, taking S⊆ ΓN, we denote as S± 1 4 the set of vertices of Γ fine N that are at a distance lesser or equal...
[44]

The family {πk 1πl 2̸= 0|k∈ K1, l∈ K2} is finite because K1 and K2 are finite, and its elements are non zero by definition. Now let πi 3 =πk 1πl 2 andπj 3 =πm 1 πn 2 be elements of {πk 1πl 2̸= 0|k∈ K1, l∈ K2}, be- cause theπk 1 andπl 2 are atomic projectors we have that (πi 3)...
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Fi- nally,∑ iπi 3 =∑ k,lπk 1πl 2 =∑ kπk 1 ∑ lπl 2 = 1. B.2 Sub-C* algebras of a centre are coarse- grainings The following lemma shows that any sub-C* alge- bra of a centre can essentially be seen as a coarse- graining, in the sense that its atomic projectors are lumping toget...
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Lemma B.3 shows thatZ(ω)∨A 2 is equal to the latter’s RHS, so A′ 1 =Z(ω)∨A 2

Yet using Lemma D.1 then the assumption (A1,A2)⊢ ω, we have for ev- ery k: πkA′ 1 = A′ 1∩ Im(ˆπk) = A′ 1∩ πkω = πkA2, so A′ 1 = ⨄ kπkA2. Lemma B.3 shows thatZ(ω)∨A 2 is equal to the latter’s RHS, so A′ 1 =Z(ω)∨A 2. Proposition B.5. If (A1,A2)⊢ ω, then Z1⊆Z (ω)∨Z 2. (57) In par...
[47]

Therefore the previous reasoning applies to it, soZJn+l+2,−1K = (Zn+l+1∩Zn+l+2)∨(Z−1∩Z0)

= N−n−l− 3≥ N− 2l− 2≥ l, where we used n≤ l− 1 then N≥ 3l + 2. Therefore the previous reasoning applies to it, soZJn+l+2,−1K = (Zn+l+1∩Zn+l+2)∨(Z−1∩Z0). The same applies toZJn+1,n+l+1K. This allows us to compute ZJ0,nK =ZJn+1,−1K ⊆Z Jn+1,n+l+1K∨Z Jn+l+2,−1K ⊆ (Zn∩Zn+1)∨ (Zn+l+...
[48]

(78) Proof

Then π(F1∩F 2)⊆πF1∩πF2. (78) Proof. An element of the LHS is of the form πf with f∈F 1∩F 2, and is therefore in particular in πF1 and in πF2. A crucial stake is our ability to go in the other direction, i.e. to factor a projection out of an intersection. Taking again π a proje...
[49]

(81) Proof

Then π(F′ 1∩F 2)⊆F′ 1∩πF2. (81) Proof. An element of the LHS is of the form πf with f∈F ′ 1∩F 2; as π∈F ′ 1, πf∈F ′ 1, so πf∈ F′ 1∩πF2. Lemma D.5. Working in a C*-algebra ω, let F1,F2⊆G be sub-C* algebras, and π be an or- thogonal projector inG′. Then F′ 1∩πF2 =π(F′ 1∩µF2), (8...
[50]

The reverse inclusion is obtained through π(F′ 1∩µF2)⊆F ′ 1∩πµF2 =F′ 1∩πF2, where we used Lemma D.4 then πµ =π

Therefore πf2 =πµf2∈π(F′ 1∩µF2), soF′ 1∩πF2⊆π(F′ 1∩µF2). The reverse inclusion is obtained through π(F′ 1∩µF2)⊆F ′ 1∩πµF2 =F′ 1∩πF2, where we used Lemma D.4 then πµ =π. Recall that in the context of a strongly con- nected 1D partition, what really matters are the ‘edge centres...
[51]

Yet ˆπi is injective on AJm,nK, since Lemma D.6 (applied with our as- sumption d({k,k + 1},{m,n})≥l), tells us that the intersection of its kernel with the latter is designated by a ¯µ = 0; so gi 1 =gi 2 for every i. Thus, taking an elementf of (83a)’s LHS, it is of the form f...
[52]

D.2 Vertex-splitting The crucial tool in the proof of Theorem 3.1 is the following refinement lemma

⊆ (Zk∩ Zk+1)∨ (F2∩F′ 1). D.2 Vertex-splitting The crucial tool in the proof of Theorem 3.1 is the following refinement lemma. It tells us which data specification is sufficient to ‘split a vertex’ in a strongly connected 1D partition of a factor, i.e. to turn it into a partiti...
[53]

We also write q9 :=p9 +l and q+:=p+−l

If N is odd, we write p9 and p+:= p9 + 1 as the two elements furthest away from 0; if N is even, we write p9 as the furthest element from 0 and p+= p9 + 1 (in other words, in both cases, p9 :=−⌈N−1 2 ⌉ and p+:=⌊N−1 2 ⌋). We also write q9 :=p9 +l and q+:=p+−l. Suppose there exi...
[54]

∀S ⊆ ΓN \ {0}, Asplit S := AS and Asplit S⊔{− 1 4, 1 4} :=AS⊔{0}
[55]

∀m ∈ Jp9,− 1 4 K, Asplit Jm,− 1 4 K := AJm,0K ∩ Asplit Jp9,− 1 4 K
[56]

∀m∈ J 1 4,p +K, Asplit J 1 4,mK :=AJ0,mK∩A split J 1 4,p+K
[57]

algebras for intervals of size larger than N 2 are defined via the rule Asplit ¯S := (Asplit S )′
[58]

Before we turn to the proof, let us make a few comments

Asplit S for S not an interval is defined as the algebraic span of the algebras for each of S’s connected components. Before we turn to the proof, let us make a few comments. First, note that since N≥ 4l + 1≥ 3l+2, (AS)S⊆ΓN is necessarily strongly connected by Theorem C.2, i.e...
[59]

First, (91a) and (91b) tell us that for any m, AJm,− 1 4 K ⊆ AJm,0K andAJ 1 4,mK ⊆ AJ0,mK; after splitting every vertex, this yields (109a)

Like before, we will de- noteAsplit S asAS, since the presence of ± 1 4 in S is sufficient to disambiguate one from the other. First, (91a) and (91b) tell us that for any m, AJm,− 1 4 K ⊆ AJm,0K andAJ 1 4,mK ⊆ AJ0,mK; after splitting every vertex, this yields (109a). To eventu...
[60]

This comes from the fact that for every S⊆ Γfine N , we have ˜AS ⊆ AS± 1 4 ⊥A S±( 1 4 +lA) ⊇ AS±( 1 2 +lA)± 1 4 ⊇ ˜AS±( 1 2 +lA). Note that in the first inclusion there, we used (109a) in the other direction; we can do this because of the invertibility of QCAs with a given cau...
[61]

Furthermore, N >4r + 2(l +l′) = 4(r− 1
[62]

+ 2((l + 1) +l′). If we are in the base case, we find ∀S ⊆ Γ+1/2 N , ˜Acoarse S = U−1(BS); so, defining Ufine 1 := I (the identity automorphism on ωin) and Ucoarse 1 :=U, and seeing the first one as mapping from (AS)S⊆ΓN to ( ˜AS)S⊆Γfine N , and the second as mapping from the ...
[2021]

013107, arXiv:2007.11905 [cond-mat.stat-mech]

2007 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [21]

Partitions in quantum theory,

A. Vanrietvelde, O. Mestoudjian, and P. Arrighi, “Partitions in quantum theory,”. Concurrently appearing on the arxiv

work page

[2] [1]

Quantum common causes and quantum causal models

J.-M. A. Allen, J. Barrett, D. C. Horsman, C. M. Lee, and R. W. Spekkens, “Quantum common causes and quantum causal models,” Physical Review X 7 (2017) 031021, arXiv:1609.09487 [quant-ph]

work page Pith review arXiv 2017

[3] [2]

Quantum causal models,

J. Barrett, R. Lorenz, and O. Oreshkov, “Quantum causal models,” arXiv:1906.10726 [quant-ph]

arXiv 1906

[4] [3]

Causal structure in the presence of sectorial constraints, with application to the quantum switch,

N. Ormrod, A. Vanrietvelde, and J. Barrett, “Causal structure in the presence of sectorial constraints, with application to the quantum switch,” Quantum 7 (June, 2023) 1028

work page 2023

[5] [4]

Semicausal operations are semilocalizable,

T. Eggeling, D. Schlingemann, and R. F. Werner, “Semicausal operations are semilocalizable,” Europhysics Letters (EPL) 57 no. 6, (2002) 782–788, arXiv:quant-ph/0104027

arXiv 2002

[6] [5]

Causal and compositional structure of unitary transformations,

R. Lorenz and J. Barrett, “Causal and compositional structure of unitary transformations,” Quantum 5 (2021) 511, arXiv:2001.07774 [quant-ph]

arXiv 2021

[7] [6]

Routed quantum circuits

A. Vanrietvelde, H. Kristj´ ansson, and J. Barrett, “Routed quantum circuits,” Quantum 5 (Jul, 2021) 503, arXiv:2011.08120 [quant-ph]

work page Pith review arXiv 2021

[8] [7]

Vanrietvelde, Routed quantum circuits: an extended framework for coherent control and indefinite causal order

A. Vanrietvelde, Routed quantum circuits: an extended framework for coherent control and indefinite causal order. PhD thesis, University of Oxford ; Imperial college London, Sept., 2022

work page 2022