Classically efficient regimes in measurement based quantum computation performed using diagonal two qubit gates and cluster measurements

Michael Garn; Sahar Atallah; Shashank Virmani; Yukuan Tao

arxiv: 2307.01800 · v2 · submitted 2023-07-04 · 🪐 quant-ph

Classically efficient regimes in measurement based quantum computation performed using diagonal two qubit gates and cluster measurements

Sahar Atallah , Michael Garn , Yukuan Tao , Shashank Virmani This is my paper

Pith reviewed 2026-05-24 07:23 UTC · model grok-4.3

classification 🪐 quant-ph

keywords measurement-based quantum computationdiagonal gatesclassical simulationcluster statesseparabilitycylindrical setsentangled statesthermal states

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The pith

Any two-qubit diagonal gate has an explicitly computable λ that marks a classically simulatable regime for measurement-based quantum computation on finite-degree graphs.

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

The paper computes the threshold parameter λ for every two-qubit diagonal gate, extending an earlier result that applied only to controlled-Z gates. This λ fixes how close each initial qudit must stay to a diagonal state, measured by a specific distance, so that destructive measurements in the computational basis or any unbiased basis remain classically efficient. On any graph whose nodes have bounded degree, the result carves out a two-parameter family of pure entangled states (and a three-parameter family of thermal states) inside which the permitted measurements can be simulated efficiently, even though other parameter values in the same family can realize ideal cluster-state computation.

Core claim

By calculating λ for arbitrary diagonal two-qubit gates the work shows that, for any finite-degree graph, there is a two-parameter family of pure entangled states (or three-parameter family of thermal states) whose permitted measurements admit efficient classical simulation, although other values of the same parameters can support ideal cluster-state quantum computation. The calculation rests on a separability criterion expressed through cylindrical sets of operators; the authors also prove optimality of this choice inside a broad class of sets while exhibiting numerically stronger sets outside that class.

What carries the argument

Explicit computation of λ for each two-qubit diagonal gate by testing separability inside cylindrical sets of operators; λ directly sets the radius of the classically efficient ball around diagonal initial states.

If this is right

The classically efficient regime applies uniformly to every finite-degree graph.
A two-parameter family of pure entangled states on such graphs remains classically simulatable under the allowed measurements.
A three-parameter family of thermal states is likewise covered by the same bound.
The permitted measurements are exactly those in the computational basis and bases unbiased to it.
Cylindrical sets are optimal inside a broad class of operator sets for establishing the separability bound.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit λ now makes it possible to rank different diagonal gates by the size of their classically efficient regime.
Numerical evidence outside the cylindrical class indicates that still larger simulatable regions may exist for some gates.
The same λ construction could be applied directly to small test graphs to verify the predicted boundary between simulable and non-simulable regimes.

Load-bearing premise

Initial qudits must lie within λ to the power of minus D of a diagonal state under the given distance, and all measurements must be performed in the computational basis or in bases unbiased to it.

What would settle it

A concrete numerical counter-example would be any two-qubit diagonal gate together with a small finite graph on which states initialized exactly at distance λ^{-D} produce measurement statistics that cannot be reproduced by the classical algorithm whose correctness the paper claims.

Figures

Figures reproduced from arXiv: 2307.01800 by Michael Garn, Sahar Atallah, Shashank Virmani, Yukuan Tao.

**Figure 2.** Figure 2: FIG. 2. The plot shows is a plot of polar angle [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. A sketch of [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

read the original abstract

In a recent work arXiv:2201.07655v2 we showed that there is a constant $\lambda >0$ such that it is possible to efficiently classically simulate a quantum system in which (i) qudits are placed on the nodes of a graph, (ii) each qudit undergoes at most $D$ diagonal gates, (iii) each qudit is destructively measured in the computational basis or bases unbiased to it, and (iv) each qudit is initialised within $\lambda^{-D}$ of a diagonal state according to a particular distance measure. In this work we explicitly compute $\lambda$ for any two qubit diagonal gate, thereby extending the computation of arXiv:2201.07655v2 beyond CZ gates. For any finite degree graph this allows us to describe a two parameter family of pure entangled quantum states (or three parameter family of thermal states) which have a non-trivial classically efficiently simulatable "phase" for the permitted measurements, even though other values of the parameters may enable ideal cluster state quantum computation. The main the technical tool involves considering separability in terms of "cylindrical" sets of operators. We also consider whether a different choice of set can strengthen the algorithm, and prove that they are optimal among a broad class of sets, but also show numerically that outside this class there are choices that can increase the size of the classically efficient regime.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They compute explicit λ for any diagonal two-qubit gate, giving concrete two- and three-parameter families of simulable states on finite-degree graphs.

read the letter

The main new result is the explicit value of λ that works for arbitrary diagonal two-qubit gates rather than only CZ. This produces two-parameter families of pure entangled states and three-parameter families of thermal states that remain classically simulable on finite-degree graphs when the initial states sit within λ^{-D} of diagonal according to their distance and the measurements stay in the computational or unbiased bases. The authors also run a numerical search that finds stronger sets outside the cylindrical class they analyze analytically, and they prove optimality inside that class. That combination of closed-form computation and numerical check is the part that stands out as useful work. The argument stays inside the separability framework from their earlier paper, so there is no circularity or hidden fitting. The derivation steps look reproducible on the basis of the abstract and the stress-test note. The soft spots are the same ones that were already present: the allowed initialization distance shrinks exponentially with degree D, the graphs must have finite degree, and the measurements are restricted. Those constraints keep the simulable regime from covering the full cluster-state case, but the paper states the limits clearly and does not overclaim. The result is incremental rather than a broad shift in the field. Specialists working on classical simulation thresholds for MBQC or on resource-state families will find the explicit families and the numerical improvements worth reading. A reader who wants tighter bounds on when diagonal-gate MBQC stays easy to simulate will get concrete value. The paper shows clear engagement with the literature and the prior framework, so it deserves a serious referee even though the advance is narrow. I would recommend sending it to peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript extends the authors' prior result (arXiv:2201.07655v2) by explicitly computing the constant λ > 0 for arbitrary two-qubit diagonal gates. In the model, qudits on a graph undergo at most D such gates, are initialized within λ^{-D} of a diagonal state under a stated distance measure, and are measured in the computational basis or unbiased bases. Using separability with respect to cylindrical sets of operators, the work identifies a two-parameter family of pure entangled states (or three-parameter family of thermal states) on finite-degree graphs that remain classically efficiently simulatable, even though other parameter values permit ideal cluster-state quantum computation. The cylindrical sets are shown to be optimal inside a broad class, with numerical evidence that superior sets exist outside it.

Significance. If the explicit λ computation holds, the result supplies concrete, gate-specific thresholds that delineate classically simulatable regimes in MBQC, thereby sharpening the boundary between efficient classical simulation and potential quantum advantage. Credit is due for the parameter-free derivation of λ via the cylindrical-set construction and for the accompanying optimality proof within the stated class together with numerical checks outside it; these features make the simulatable phase directly usable for any diagonal gate.

minor comments (2)

[Abstract] Abstract: the sentence 'The main the technical tool involves...' contains a duplicated word and should read 'The main technical tool involves...'.
[§2 (or equivalent)] The distance measure used for the λ^{-D} initialization bound is referenced to the prior work but its explicit form and triangle-inequality properties are not restated; a brief self-contained definition in §2 or an appendix would improve readability without altering the central argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript, the recognition of its significance in providing explicit, gate-specific thresholds for classical simulability in MBQC, and the recommendation of minor revision. The referee's description accurately reflects the extension of our prior work to arbitrary diagonal two-qubit gates via the cylindrical-set construction.

Circularity Check

0 steps flagged

Minor self-citation of framework; explicit λ computation independent

full rationale

The paper cites arXiv:2201.07655v2 solely for the existence of some λ>0 and the overall simulation framework (initialization bound, permitted measurements, graph structure). The central new contribution—an explicit computation of λ for arbitrary two-qubit diagonal gates via separability with respect to cylindrical operator sets—is presented as a direct calculation that does not reduce to any fitted parameter or equation taken from the cited work. The manuscript further proves optimality of the chosen sets inside a stated broad class and supplies numerical evidence of better sets outside that class, confirming that the size of the classically efficient regime is not forced by prior self-citations. No self-definitional step, fitted-input prediction, or load-bearing uniqueness theorem appears in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the prior existence result for λ and the definition of the distance measure to diagonal states; no new free parameters are introduced, no invented entities appear, and axioms are standard mathematical definitions of separability and operator sets.

axioms (2)

domain assumption Existence of a constant λ > 0 such that systems with initial states within λ^{-D} of diagonal states admit efficient classical simulation (from arXiv:2201.07655v2)
Invoked in the abstract as the foundation being extended.
standard math Standard definitions of separability for sets of operators on qudits
Used as the main technical tool for cylindrical sets.

pith-pipeline@v0.9.0 · 5794 in / 1504 out tokens · 43683 ms · 2026-05-24T07:23:06.962465+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

66 extracted references · 66 canonical work pages · 3 internal anchors

[1]

[”,“]”, reserving curved brackets “(

(roughly speaking the idea that entanglement may be considered relative to a privileged set of observables) may be used to develop eﬃcient classical algorithms for classes of entangled quantum system [4, 13–17] that are not amenable to other methods. We will primarily build upon the work of [4], in which we developed a generalised separability based algor...

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[2]

Hence w.l.o.g

where ϕ = ϕ 4 +ϕ 1 − ϕ 2 − ϕ 3. Hence w.l.o.g. we may consider two qubit diagonal unitaries of this controlled-phase form, so that ϕ is the only free parameter. ■ As local diagonal unitaries do not aﬀect our arguments, we will consider unitaries of the form ( 3). In the case of qubits, we deﬁne a ‘cylinder of radius r’ as the following set of normalised (...

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[3]

Proof: Our goal is to establish whether the output is Cyl(RA), Cyl(RB)-separable

We refer to an operator of the form of equation ( 5) as being Cyl(RA), Cyl(RB)-separable. Proof: Our goal is to establish whether the output is Cyl(RA), Cyl(RB)-separable. Forϕ = 0 the entangling gate Vϕ is the identity and the output is trivially separable as long as fA,f B ≤ 1. So in the remainder of the proof we only consider ϕ ̸= 0. As Vϕ is a linear ...

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[4]

This shows that the decom- position (7) is invariant up to local Z-rotations, and hence we may restrict our attention to inputs to the form [1,r A, 0, ± 1] ⊗ [1,r B, 0, ± 1]

Since UA ⊗ UB commutes with Vϕ and cylin- ders are invariant under Z rotations, we have the Cyl(rA), Cyl(rB)-separable decomposition: Vϕ ( UAρAU † A ⊗ UBρBU † B ) V † ϕ = ∑ i piUAω i AU † A ⊗ UBω i BU † B where ω i k ∈ Cyl(Rk). This shows that the decom- position (7) is invariant up to local Z-rotations, and hence we may restrict our attention to inputs t...

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[5]

DeﬁningUϕ := exp(−iϕ/ 2)|0⟩⟨0|+ exp(iϕ/ 2)|1⟩⟨1|, we have the identity Vϕ (X ⊗ X) = (X ⊗ UϕX)Vϕ

If both inputs havez = − 1, then the input extrema: [1,r A, 0, − 1] ⊗ [1,r B, 0, − 1] can be expressed as X ⊗ X ([1,r A, 0, 1] ⊗ [1,r B, 0, 1])X † ⊗ X †. DeﬁningUϕ := exp(−iϕ/ 2)|0⟩⟨0|+ exp(iϕ/ 2)|1⟩⟨1|, we have the identity Vϕ (X ⊗ X) = (X ⊗ UϕX)Vϕ. Applying this to ( 7), we arrive at: Vϕ ( Xρ AX † ⊗ Xρ BX †) V † ϕ = ∑ i piXω i AX † ⊗ UϕXω i B(UϕX)† 5 Th...

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[6]

Next, consider input states of the form [1,r A, 0, 1] ⊗ [1,r B, 0, − 1] and [1,r A, 0, − 1] ⊗ [1,r B, 0, 1]. As Vϕ is symmetric between the two input states, we can switch the control and target state, there- fore we need only consider one input extremum where the z-components are are not the same

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[7]

At this stage, we have managed to reduce the inputs we need to consider to just two inputs [1,r A, 0, 1] ⊗ [1,r B, 0, ± 1]. However it will be convenient to note that because of the identity Vϕ(I ⊗ X) = eiϕ/2(Uϕ ⊗ X)V−ϕ, checking the separability of the output correspond- ing to [1 ,r A, 0, 1] ⊗ [1,r B, 0, − 1] can instead be achieved by considering the o...

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[8]

To determine this we need to determine whether it can be expressed in the form: ∑ i pi    1 RA cos(θi) RA sin(θi) 1    [ 1 RB cos(φ i) RB sin(φ i) 1 ]

can be expressed in the Pauli basis as (the quickest way to com- pute this is to note that terms X ⊗ (I +Z), (I +Z)⊗ I,Z ⊗ Z,I ⊗Z,Z ⊗I,I ⊗I are all invariant under transformation byVϕ, so one only needs to compute Vϕ(X ⊗ X)V † ϕ ):    1 rB 0 1 rA rArB(cos (ϕ ) + 1)/ 2 rArB sin(ϕ )/ 2 rA 0 rArB sin(ϕ )/ 2 rArB(1 − cos (ϕ ))/ 2 0 1 rB 0 1    Our task ...

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[9]

is Cyl (1) , Cyl (1)-separable. Following similar reasoning to [4], we will now see that determining whether equation ( 8) is Cyl(1), Cyl(1)- separable is equivalent to checking quantum separability of a diﬀerent two-qubit operator. First, suppose that we have a quantum separable decomposition ∑ i pi[1,x i A,y i A,z i A] ⊗ [1,x i B,y i B,z i B] for the fo...

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[10]

Conversely, suppose now that this oper- ator has a Cyl(1) , Cyl(1)-separable decomposition given by ( 11)

by setting the coeﬃcients corresponding to at least one Z operator to zero):    1 fB 0 0 fA fAfB (cos (ϕ ) + 1)/ 2 fAfB sin (ϕ )/ 2 0 0 fAfB sin (ϕ )/ 2 fAfB (1 − cos (ϕ ))/ 2 0 0 0 0 0    (10) Now given such a decomposition, then by taking this de- composition and resetting zi A =zi B = 1 for all i, it follows that ∑ i pi[1,x i A,y i A, 1] ⊗ [1,x i...

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[11]

In order for it to be quantum separable, it must be both positive and a PPT (‘positive partial transpose’ [6]) op- erator

is quantum separable. In order for it to be quantum separable, it must be both positive and a PPT (‘positive partial transpose’ [6]) op- erator. However for operators of the form of (

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[12]

This means that the operator ( 10) and its partial trans- position have identical eigenvalues, and we merely need to check that one of them is positive semi-deﬁnite

the partial transpose with respect to particle A can simply be implemented by applying an X ⊗ I transformation. This means that the operator ( 10) and its partial trans- position have identical eigenvalues, and we merely need to check that one of them is positive semi-deﬁnite. It is convenient to re-express (

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[13]

To simplify notation we set fAfB(eiϕ − 1) = ceiγ, where c> 0 can be taken to be positive as by assumption ϕ ̸= 0

as: (I +fAX) ⊗ (I +fBX) +fAfB(e−iϕ − 1)|00⟩⟨11| +fAfB(eiϕ − 1)|11⟩⟨00|. To simplify notation we set fAfB(eiϕ − 1) = ceiγ, where c> 0 can be taken to be positive as by assumption ϕ ̸= 0. We also note that this implies that eiϕ − 1 has a strictly negative real component, and hence c cos(γ)< 0. Hence (10) becomes: (I +fAX) ⊗ (I +fBX) +c ( e−iγ|00⟩⟨11|+eiγ|11...

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[14]

In this case it is impossible to be separable, because (for example) from equa- tion (

fA> 1 and/or fB > 1. In this case it is impossible to be separable, because (for example) from equa- tion (

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[15]

we see that the expected value of X ⊗ I or I ⊗ X would be greater than 1, which is impossible for a separable quantum state

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[16]

Assume that fA = 1 (if instead fB = 1 the argument is identical by sym- metry)

fA = 1 and/or fB = 1. Assume that fA = 1 (if instead fB = 1 the argument is identical by sym- metry). This means that the term ( I +fAX) ⊗ (I +fBX) has at least two zero eigenvalues, with eigenstates | − −⟩ and | − +⟩. Computing the ﬁrst of these overlaps gives: ⟨− − | c ( eiγ|00⟩⟨11|+ce−iγ|11⟩⟨00| ) | − −⟩ = c 2 cos(γ). Under our assumption that ϕ ̸= 0 t...

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The ﬁnal possibility is fA,f B < 1. Let us deﬁne two operators K := (I +fAX) ⊗ (I +fBX) +c (|00⟩⟨00|+ |11⟩⟨11|) and L := −c ( |00⟩⟨00|+ |11⟩⟨11| −e−iγ|00⟩⟨11| −eiγ|11⟩⟨00| ) Then we have that equation ( 12) can be re- expressed as: K +L as this corresponds to simply adding and subtract- ing c (|00⟩⟨00|+ |11⟩⟨11|) to equation ( 12). We now note that K is a...

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Hence, under the assumption that fA,f B < 1, we have a separable decomposition iﬀ the determinant of (

can have at most one nega- tive eigenvalue. Hence, under the assumption that fA,f B < 1, we have a separable decomposition iﬀ the determinant of (

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As the de- terminant is given by equation ( 6) this establishes the lemma

is non-negative. As the de- terminant is given by equation ( 6) this establishes the lemma. V. CLASSICALLY EFFICIENT PHASES, WITH CLUSTER STATE MEASUREMENTS We now discuss how the above lemma can be used to determine an eﬃciently classically simulatable ‘phase’ for a family of systems with two parameters r ∈ [0, 1],ϕ ∈ [0, 2π ). In particular, suppose tha...

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states with Bloch vectors [x,y,z ] such that x2 +y2 ≤ r2 - are placed at the nodes of a graph

Qubits drawn from Cyl( r) - i.e. states with Bloch vectors [x,y,z ] such that x2 +y2 ≤ r2 - are placed at the nodes of a graph

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Each qubit undergoes at most D gates Vϕ

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Each qubit is then measured adaptively in the com- putational basis or equatorial bases. We will apply the lemma, following the approach of [4], to derive a region of r,ϕ for which the system can be eﬃciently simulated classically, in the sense of sampling to within an arbitrary ﬁxed total variational distance in polynomial time. We begin by considering t...

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in the sym- metric case fA = fB = f . In this case the determinant simpliﬁes to: ( 1 − f 2) ( ( 1 − f 2) 3 + 4(cos(ϕ ) − 1)f 4 ) As 1 − f 2 is always positive (apart from f = 1, which we are excluding when ϕ ̸= 0), we have separability if and only if: ( 1 − f 2) 3 + 4(cos(ϕ ) − 1)f 4 ≥ 0 In terms of the ratios R/r this can be written as (also multiplying ...

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We have that α< 0 as we are only considering ϕ ̸= 0, and we have that t> 0 as we needR/r> 1

a little, let us set α := 4(cos(ϕ ) − 1) and t = R2 r2 − 1. We have that α< 0 as we are only considering ϕ ̸= 0, and we have that t> 0 as we needR/r> 1. Hence we are looking for solutions to the cubic t3 +αt +α ≥ 0 (15) in the region t> 0. The part t3 +αt is an odd function of t with a root at t = 0, and it satisﬁes t3 +αt → ∞ as t → ∞ . As α is negative,...

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would then become { (ϕ,θ,T ) ⏐ ⏐ ⏐ ⏐θ ≤ sin−1 ( λ(ϕ )−D 1 − 2pT )} Local dephasing noise can be handled in a similar way - it simply changes the eﬀective size of the starting cylinder that we can eﬃciently simulate, making it larger. We note that while incorporating noise/temperature in this way is straightforward, we anticipate that other methods such as...

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In the previous classical simulation we found that two elements of Cyl(r), after a CZ gate, would be separable w.r.t

The initial state |ψ ⟩ is con- tained within B(r, √ 1 − r2) ⊂ B(r, 1) = Cyl( r). In the previous classical simulation we found that two elements of Cyl(r), after a CZ gate, would be separable w.r.t. a bigger cylinder Cyl(λ(π )r). However, we have found nu- merically for most values of r that when two elements of B(r, √ 1 − r2) undergo a CZ gate, then they...

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Jozsa, R. On the simulation of quantum circuits. ArXiv Preprint quant-ph/0603163. (2006)

work page internal anchor Pith review Pith/arXiv arXiv 2006
[56]

Kissinger, J

A. Kissinger, J. van de Wetering, Universal MBQC with generalised parity-phase interactions and Pauli measure- ments. Quantum 3, 134 (2019)

work page 2019
[57]

Miyake, Quantum Computation on the Edge of a Symmetry-Protected Topological Order

A. Miyake, Quantum Computation on the Edge of a Symmetry-Protected Topological Order. Phys. Rev. Lett. 105, 040501 (2010)

work page 2010
[58]

Miller and A

J. Miller and A. Miyake, Resource Quality of a Symmetry-Protected Topologically Ordered Phase for Quantum Computation. Phys. Rev. Lett. 114, 120506 (2015)

work page 2015
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D. V. Else, I. Schwarz, S. D. Bartlett, and A. C. Doherty, Symmetry-Protected Phases for Measurement- Based Quantum Computation. Phys. Rev. Lett. 108, 240505 (2012)

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[60]

Raussendorf, A

R. Raussendorf, A. Prakash, D. S. Wang, T. C.Wei, D. T. Stephen, Symmetry-protected topological phases with uniform computational power in one dimension. Phys. Rev. A 96, 012302 (2017)

work page 2017
[61]

Raussendorf, C

R. Raussendorf, C. Okay, D. S. Wang, D. T. Stephen, and H. Poulsen Nautrup, Computationally Universal Phase of Quantum Matter. Phys. Rev. Lett. 122, 090501 (2019)

work page 2019
[62]

Harrington, and K

R Raussendorf, J. Harrington, and K. Goyal, Topologi- cal fault-tolerance in cluster state quantum computation. New J. Phys. 9 199 (2007)

work page 2007
[63]

Now replace each initial qubit |+⟩ state with a ‘bag’ of n physical qubits each prepared in |ψ⟩ := cos( θ/2)|0⟩ + sin( θ/2)|1⟩

Consider a fault tolerant construction of cluster state quantum computation, such as [36]. Now replace each initial qubit |+⟩ state with a ‘bag’ of n physical qubits each prepared in |ψ⟩ := cos( θ/2)|0⟩ + sin( θ/2)|1⟩. The state |ψ⟩⊗n can be expressed as a sum of even and odd parity states, with amplitudes that equalise in magni- tude as n is increased. N...

work page
[64]

Walgate, A

J. Walgate, A. J. Short, L. Hardy, and V. Vedral, Local Distinguishability of Multipartite Orthogonal Quantum States. Phys. Rev. Lett. 85, 4972 (2000)

work page 2000
[65]

Virmani, M

S. Virmani, M. F. Sacchi, M. B. Plenio, D. Markham. Op- timal local discrimination of two multipartite pure states , Phys. Lett. A. 288 62 (2001)

work page 2001
[66]

M. Garn, Y. Tao, and S. Virmani, in preparation

work page

[1] [1]

[”,“]”, reserving curved brackets “(

(roughly speaking the idea that entanglement may be considered relative to a privileged set of observables) may be used to develop eﬃcient classical algorithms for classes of entangled quantum system [4, 13–17] that are not amenable to other methods. We will primarily build upon the work of [4], in which we developed a generalised separability based algor...

work page

[2] [2]

Hence w.l.o.g

where ϕ = ϕ 4 +ϕ 1 − ϕ 2 − ϕ 3. Hence w.l.o.g. we may consider two qubit diagonal unitaries of this controlled-phase form, so that ϕ is the only free parameter. ■ As local diagonal unitaries do not aﬀect our arguments, we will consider unitaries of the form ( 3). In the case of qubits, we deﬁne a ‘cylinder of radius r’ as the following set of normalised (...

work page

[3] [3]

Proof: Our goal is to establish whether the output is Cyl(RA), Cyl(RB)-separable

We refer to an operator of the form of equation ( 5) as being Cyl(RA), Cyl(RB)-separable. Proof: Our goal is to establish whether the output is Cyl(RA), Cyl(RB)-separable. Forϕ = 0 the entangling gate Vϕ is the identity and the output is trivially separable as long as fA,f B ≤ 1. So in the remainder of the proof we only consider ϕ ̸= 0. As Vϕ is a linear ...

work page

[4] [4]

This shows that the decom- position (7) is invariant up to local Z-rotations, and hence we may restrict our attention to inputs to the form [1,r A, 0, ± 1] ⊗ [1,r B, 0, ± 1]

Since UA ⊗ UB commutes with Vϕ and cylin- ders are invariant under Z rotations, we have the Cyl(rA), Cyl(rB)-separable decomposition: Vϕ ( UAρAU † A ⊗ UBρBU † B ) V † ϕ = ∑ i piUAω i AU † A ⊗ UBω i BU † B where ω i k ∈ Cyl(Rk). This shows that the decom- position (7) is invariant up to local Z-rotations, and hence we may restrict our attention to inputs t...

work page

[5] [5]

DeﬁningUϕ := exp(−iϕ/ 2)|0⟩⟨0|+ exp(iϕ/ 2)|1⟩⟨1|, we have the identity Vϕ (X ⊗ X) = (X ⊗ UϕX)Vϕ

If both inputs havez = − 1, then the input extrema: [1,r A, 0, − 1] ⊗ [1,r B, 0, − 1] can be expressed as X ⊗ X ([1,r A, 0, 1] ⊗ [1,r B, 0, 1])X † ⊗ X †. DeﬁningUϕ := exp(−iϕ/ 2)|0⟩⟨0|+ exp(iϕ/ 2)|1⟩⟨1|, we have the identity Vϕ (X ⊗ X) = (X ⊗ UϕX)Vϕ. Applying this to ( 7), we arrive at: Vϕ ( Xρ AX † ⊗ Xρ BX †) V † ϕ = ∑ i piXω i AX † ⊗ UϕXω i B(UϕX)† 5 Th...

work page

[6] [6]

Next, consider input states of the form [1,r A, 0, 1] ⊗ [1,r B, 0, − 1] and [1,r A, 0, − 1] ⊗ [1,r B, 0, 1]. As Vϕ is symmetric between the two input states, we can switch the control and target state, there- fore we need only consider one input extremum where the z-components are are not the same

work page

[7] [7]

At this stage, we have managed to reduce the inputs we need to consider to just two inputs [1,r A, 0, 1] ⊗ [1,r B, 0, ± 1]. However it will be convenient to note that because of the identity Vϕ(I ⊗ X) = eiϕ/2(Uϕ ⊗ X)V−ϕ, checking the separability of the output correspond- ing to [1 ,r A, 0, 1] ⊗ [1,r B, 0, − 1] can instead be achieved by considering the o...

work page

[8] [8]

To determine this we need to determine whether it can be expressed in the form: ∑ i pi    1 RA cos(θi) RA sin(θi) 1    [ 1 RB cos(φ i) RB sin(φ i) 1 ]

can be expressed in the Pauli basis as (the quickest way to com- pute this is to note that terms X ⊗ (I +Z), (I +Z)⊗ I,Z ⊗ Z,I ⊗Z,Z ⊗I,I ⊗I are all invariant under transformation byVϕ, so one only needs to compute Vϕ(X ⊗ X)V † ϕ ):    1 rB 0 1 rA rArB(cos (ϕ ) + 1)/ 2 rArB sin(ϕ )/ 2 rA 0 rArB sin(ϕ )/ 2 rArB(1 − cos (ϕ ))/ 2 0 1 rB 0 1    Our task ...

work page

[9] [9]

is Cyl (1) , Cyl (1)-separable. Following similar reasoning to [4], we will now see that determining whether equation ( 8) is Cyl(1), Cyl(1)- separable is equivalent to checking quantum separability of a diﬀerent two-qubit operator. First, suppose that we have a quantum separable decomposition ∑ i pi[1,x i A,y i A,z i A] ⊗ [1,x i B,y i B,z i B] for the fo...

work page

[10] [10]

Conversely, suppose now that this oper- ator has a Cyl(1) , Cyl(1)-separable decomposition given by ( 11)

by setting the coeﬃcients corresponding to at least one Z operator to zero):    1 fB 0 0 fA fAfB (cos (ϕ ) + 1)/ 2 fAfB sin (ϕ )/ 2 0 0 fAfB sin (ϕ )/ 2 fAfB (1 − cos (ϕ ))/ 2 0 0 0 0 0    (10) Now given such a decomposition, then by taking this de- composition and resetting zi A =zi B = 1 for all i, it follows that ∑ i pi[1,x i A,y i A, 1] ⊗ [1,x i...

work page

[11] [11]

In order for it to be quantum separable, it must be both positive and a PPT (‘positive partial transpose’ [6]) op- erator

is quantum separable. In order for it to be quantum separable, it must be both positive and a PPT (‘positive partial transpose’ [6]) op- erator. However for operators of the form of (

work page

[12] [12]

This means that the operator ( 10) and its partial trans- position have identical eigenvalues, and we merely need to check that one of them is positive semi-deﬁnite

the partial transpose with respect to particle A can simply be implemented by applying an X ⊗ I transformation. This means that the operator ( 10) and its partial trans- position have identical eigenvalues, and we merely need to check that one of them is positive semi-deﬁnite. It is convenient to re-express (

work page

[13] [13]

To simplify notation we set fAfB(eiϕ − 1) = ceiγ, where c> 0 can be taken to be positive as by assumption ϕ ̸= 0

as: (I +fAX) ⊗ (I +fBX) +fAfB(e−iϕ − 1)|00⟩⟨11| +fAfB(eiϕ − 1)|11⟩⟨00|. To simplify notation we set fAfB(eiϕ − 1) = ceiγ, where c> 0 can be taken to be positive as by assumption ϕ ̸= 0. We also note that this implies that eiϕ − 1 has a strictly negative real component, and hence c cos(γ)< 0. Hence (10) becomes: (I +fAX) ⊗ (I +fBX) +c ( e−iγ|00⟩⟨11|+eiγ|11...

work page

[14] [14]

In this case it is impossible to be separable, because (for example) from equa- tion (

fA> 1 and/or fB > 1. In this case it is impossible to be separable, because (for example) from equa- tion (

work page

[15] [15]

we see that the expected value of X ⊗ I or I ⊗ X would be greater than 1, which is impossible for a separable quantum state

work page

[16] [16]

Assume that fA = 1 (if instead fB = 1 the argument is identical by sym- metry)

fA = 1 and/or fB = 1. Assume that fA = 1 (if instead fB = 1 the argument is identical by sym- metry). This means that the term ( I +fAX) ⊗ (I +fBX) has at least two zero eigenvalues, with eigenstates | − −⟩ and | − +⟩. Computing the ﬁrst of these overlaps gives: ⟨− − | c ( eiγ|00⟩⟨11|+ce−iγ|11⟩⟨00| ) | − −⟩ = c 2 cos(γ). Under our assumption that ϕ ̸= 0 t...

work page

[17] [17]

The ﬁnal possibility is fA,f B < 1. Let us deﬁne two operators K := (I +fAX) ⊗ (I +fBX) +c (|00⟩⟨00|+ |11⟩⟨11|) and L := −c ( |00⟩⟨00|+ |11⟩⟨11| −e−iγ|00⟩⟨11| −eiγ|11⟩⟨00| ) Then we have that equation ( 12) can be re- expressed as: K +L as this corresponds to simply adding and subtract- ing c (|00⟩⟨00|+ |11⟩⟨11|) to equation ( 12). We now note that K is a...

work page

[18] [18]

Hence, under the assumption that fA,f B < 1, we have a separable decomposition iﬀ the determinant of (

can have at most one nega- tive eigenvalue. Hence, under the assumption that fA,f B < 1, we have a separable decomposition iﬀ the determinant of (

work page

[19] [19]

As the de- terminant is given by equation ( 6) this establishes the lemma

is non-negative. As the de- terminant is given by equation ( 6) this establishes the lemma. V. CLASSICALLY EFFICIENT PHASES, WITH CLUSTER STATE MEASUREMENTS We now discuss how the above lemma can be used to determine an eﬃciently classically simulatable ‘phase’ for a family of systems with two parameters r ∈ [0, 1],ϕ ∈ [0, 2π ). In particular, suppose tha...

work page

[20] [20]

states with Bloch vectors [x,y,z ] such that x2 +y2 ≤ r2 - are placed at the nodes of a graph

Qubits drawn from Cyl( r) - i.e. states with Bloch vectors [x,y,z ] such that x2 +y2 ≤ r2 - are placed at the nodes of a graph

work page

[21] [21]

Each qubit undergoes at most D gates Vϕ

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[22] [22]

Each qubit is then measured adaptively in the com- putational basis or equatorial bases. We will apply the lemma, following the approach of [4], to derive a region of r,ϕ for which the system can be eﬃciently simulated classically, in the sense of sampling to within an arbitrary ﬁxed total variational distance in polynomial time. We begin by considering t...

work page

[23] [23]

in the sym- metric case fA = fB = f . In this case the determinant simpliﬁes to: ( 1 − f 2) ( ( 1 − f 2) 3 + 4(cos(ϕ ) − 1)f 4 ) As 1 − f 2 is always positive (apart from f = 1, which we are excluding when ϕ ̸= 0), we have separability if and only if: ( 1 − f 2) 3 + 4(cos(ϕ ) − 1)f 4 ≥ 0 In terms of the ratios R/r this can be written as (also multiplying ...

work page

[24] [24]

We have that α< 0 as we are only considering ϕ ̸= 0, and we have that t> 0 as we needR/r> 1

a little, let us set α := 4(cos(ϕ ) − 1) and t = R2 r2 − 1. We have that α< 0 as we are only considering ϕ ̸= 0, and we have that t> 0 as we needR/r> 1. Hence we are looking for solutions to the cubic t3 +αt +α ≥ 0 (15) in the region t> 0. The part t3 +αt is an odd function of t with a root at t = 0, and it satisﬁes t3 +αt → ∞ as t → ∞ . As α is negative,...

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[25] [25]

would then become { (ϕ,θ,T ) ⏐ ⏐ ⏐ ⏐θ ≤ sin−1 ( λ(ϕ )−D 1 − 2pT )} Local dephasing noise can be handled in a similar way - it simply changes the eﬀective size of the starting cylinder that we can eﬃciently simulate, making it larger. We note that while incorporating noise/temperature in this way is straightforward, we anticipate that other methods such as...

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In the previous classical simulation we found that two elements of Cyl(r), after a CZ gate, would be separable w.r.t

The initial state |ψ ⟩ is con- tained within B(r, √ 1 − r2) ⊂ B(r, 1) = Cyl( r). In the previous classical simulation we found that two elements of Cyl(r), after a CZ gate, would be separable w.r.t. a bigger cylinder Cyl(λ(π )r). However, we have found nu- merically for most values of r that when two elements of B(r, √ 1 − r2) undergo a CZ gate, then they...

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R. Raussendorf and H.J. Briegel, A One-Way Quantum Computer. Phys. Rev. Lett. 86, 5188 (2001)

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S. Barrett, S. Bartlett, A. Doherty, D. Jennings, D. & T. Rudolph, Transitions in the computational power of thermal states for measurement-based quantum compu- tation. Physical Review A . 80, 062328 (2009)

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Browne, D., Elliott, M., Flammia, S., Merkel, S., Miyake, A. & Short, A. Phase transition of computational power in the resource states for one-way quantum computation. New Journal Of Physics . 10, 023010 (2008)

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S. Atallah, M. Garn, S. Jevtic, Y. Tao, and S. Virmani, Eﬃcient classical simulation of cluster state quantum cir- cuits with alternative inputs, arXiv:2201.07655

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M. Van den Nest, A. Miyake, W. D¨ ur, H. J. Briegel, Uni- versal Resources for Measurement-Based Quantum Com- putation. Phys. Rev. Lett. 97, 150504 (2006)

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D. Gross and J. Eisert, Novel Schemes for Measurement- Based Quantum Computation. Phys. Rev. Lett. 98, 220503 (2007)

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H. Barnum, E. Knill, G. Ortiz, and L. Viola, General- izations of entanglement based on coherent states and convex sets. Phys. Rev. A 68, 032308 (2003)

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R. Somma, H. Barnum, G. Ortiz, and E. Knill, Eﬃcient Solvability of Hamiltonians and Limits on the Power of Some Quantum Computational Models. Phys. Rev. Lett. 97, 190501 (2006)

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N. Ratanje and S. Virmani, Generalized state spaces and nonlocality in fault-tolerant quantum-computing schemes. Phys. Rev. A 83 032309 (2011)

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H. Anwar, S Jevtic, O. Rudolph, and S. Virmani, Families of pure PEPS with eﬃciently simulatable lo- cal hidden variable models for most measurements. arXiv:1412.3780v2

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Jozsa, R. On the simulation of quantum circuits. ArXiv Preprint quant-ph/0603163. (2006)

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A. Kissinger, J. van de Wetering, Universal MBQC with generalised parity-phase interactions and Pauli measure- ments. Quantum 3, 134 (2019)

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A. Miyake, Quantum Computation on the Edge of a Symmetry-Protected Topological Order. Phys. Rev. Lett. 105, 040501 (2010)

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J. Miller and A. Miyake, Resource Quality of a Symmetry-Protected Topologically Ordered Phase for Quantum Computation. Phys. Rev. Lett. 114, 120506 (2015)

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D. V. Else, I. Schwarz, S. D. Bartlett, and A. C. Doherty, Symmetry-Protected Phases for Measurement- Based Quantum Computation. Phys. Rev. Lett. 108, 240505 (2012)

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Raussendorf, A

R. Raussendorf, A. Prakash, D. S. Wang, T. C.Wei, D. T. Stephen, Symmetry-protected topological phases with uniform computational power in one dimension. Phys. Rev. A 96, 012302 (2017)

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Raussendorf, C

R. Raussendorf, C. Okay, D. S. Wang, D. T. Stephen, and H. Poulsen Nautrup, Computationally Universal Phase of Quantum Matter. Phys. Rev. Lett. 122, 090501 (2019)

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Harrington, and K

R Raussendorf, J. Harrington, and K. Goyal, Topologi- cal fault-tolerance in cluster state quantum computation. New J. Phys. 9 199 (2007)

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Now replace each initial qubit |+⟩ state with a ‘bag’ of n physical qubits each prepared in |ψ⟩ := cos( θ/2)|0⟩ + sin( θ/2)|1⟩

Consider a fault tolerant construction of cluster state quantum computation, such as [36]. Now replace each initial qubit |+⟩ state with a ‘bag’ of n physical qubits each prepared in |ψ⟩ := cos( θ/2)|0⟩ + sin( θ/2)|1⟩. The state |ψ⟩⊗n can be expressed as a sum of even and odd parity states, with amplitudes that equalise in magni- tude as n is increased. N...

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Walgate, A

J. Walgate, A. J. Short, L. Hardy, and V. Vedral, Local Distinguishability of Multipartite Orthogonal Quantum States. Phys. Rev. Lett. 85, 4972 (2000)

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[65] [65]

Virmani, M

S. Virmani, M. F. Sacchi, M. B. Plenio, D. Markham. Op- timal local discrimination of two multipartite pure states , Phys. Lett. A. 288 62 (2001)

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M. Garn, Y. Tao, and S. Virmani, in preparation

work page