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History-state Hamiltonians are critical
[article]

2018
*
arXiv
*
pre-print

Pr U ∼ν

arXiv:1810.06528v1
fatcat:kxfdeg7myjgh7brkd6pc2iwmvm
*s*|f h,W (U )| ≥ δ 2 ≤ 8 6144*s*d n δ 2*s*/2 . ... ) that Pr U ∼ν*s*|Re(f h,W (U ))| ≥ δ √ 2 2 ≤ 1 (δ/2 √ 2)*s*2 768s d n*s*/2 + 2ǫ(2d 2n )*s*= 4 6144s d n δ 2*s*/2 . ...##
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Uncomputably Complex Renormalisation Group Flows
[article]

2021
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arXiv
*
pre-print

We give a mathematically rigorous construction of the block-renormalisation-group map for the original undecidable many-body system that appeared in the literature (

arXiv:2102.05145v1
fatcat:pemuiuogxzgv3oatda3oel4dku
*Cubitt*, P\'erez-Garcia, Wolf, Nature ... That is, the head(*s*) of the arrow(*s*) in one tile and the tail(*s*) of the 4 cross tiles arrow(*s*) in the other tile must match exactly on the edges put into contact. × × × × × × × × × 6 up tiles × × × × × ... λ 0 (H|*S*1 ) 01 ≥ Evolve-to-Illegal Subspace,*S*2 : All standard basis states |x ∈*S*2 ⊂ B ⊗L will evolve either forwards or backwards in time to an illegal state in O(L 2 ) steps under the transition ...##
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Uncomputability of Phase Diagrams
[article]

2019
*
arXiv
*
pre-print

This work builds off recent results from

arXiv:1910.01631v1
fatcat:azmuyephnbhy3gy7sxttcurn4q
*Cubitt*et al. and Bausch et al., proving undecidability of the spectral gap problem. ... Hence for*Cubitt*, Perez-Garcia, and Wolf, = 0 if the UTM halted on its input. ... For the second claim, by the same process we thus get λ min (H*s*| A ) = E edge (*s*) + E pen,non-halt (*s*) ≥ 0. Then for the third claim, λ min (H*s*| A ) = E edge (*s*) + E pen,halt (*s*) < 0. ...##
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Entanglement in the stabilizer formalism
[article]

2004
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arXiv
*
pre-print

We further define the local subgroup

arXiv:quant-ph/0406168v1
fatcat:pjxw4yea2zaajdz6lpsnvqizr4
*S*loc of*S*as*S*loc ≡ j*S*j . (12) In the bipartite case,*S*loc =*S*A ·*S*B . ... Assuming the list is completed with e AB ≡ n − |*S*A | − |*S*B | operators generating the subgroup*S*AB , then,*S*can be decomposed as*S*=*S*A ·*S*B ·*S*AB . Note that |P A (*S*)| = n − |*S*B |. ...##
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Universal Refocusing of Systematic Quantum Noise
[article]

2016
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arXiv
*
pre-print

holds for all

arXiv:1602.07963v1
fatcat:bhk6wy4m2jhefd57ndj6ivifhm
*s*, and the lower bound holds for*s*≤ π/4. ... Thus Lemma 4 in [20] provides the result 4 10*s*≤ exp(*s*) − 1 ≤*s*• if ν ≤ π/4, then vol(exp(B ν )) ≥ 4 10 vol(B ν ) = 4 10 cν d 2 −1 ; and A, B = Tr(A † B) [19] 2 [a, b] = a T Ωb, where ...##
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Computational Complexity of the Ground State Energy Density Problem
[article]

2021
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arXiv
*
pre-print

Lockhart,

arXiv:2107.05060v1
fatcat:5nvhdl6f2ndsvorirbss75t2vy
*S*. Gharibian, and J. Sullivan. J.D.W. is supported by the EPSRC Centre for Doctoral Training in Delivering Quantum Technologies (grant EP/L015242/1). ... the total number of complete n-squares of T within*S*, and the number of complete n-squares of R within*S*. ... ( Lemma 44 Let D be a defect set in a region*S*, G = (D, E) its defect graph, and R a Robinson tiling of*S*. ...##
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Are problems in Quantum Information Theory (un)decidable?
[article]

2011
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arXiv
*
pre-print

Given a Boolean combi- nation φ(X, A) of the F i '

arXiv:1111.5425v1
fatcat:tivo3fmt3new3neekoe43fs76y
*s*and Ψ(A) := Q 1 (X 1 ∈*S*1 ) . . . ... Q n (X n ∈*S*n ) : φ(X, A) , (5) where Q j ∈ {∃, ∀} and the*S*i '*s*are semialgebraic sets (e.g. those specified in Lemma 1). Then there exists an effective algorithm which decides Ψ(A). ...##
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Comment on "On the uncomputability of the spectral gap"
[article]

2016
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arXiv
*
pre-print

*Cubitt*, D. Perez-Garcia, M.M. Wolf, Undecidability of the spectral gap, Nature 528, 207-211 (2015), arXiv:1502.04573] and his 1994 paper [

*S*. ... The aim of this short note is to clarify some of the claims made in the comparison made in [

*S*. Lloyd, On the uncomputability of the spectral gap, arXiv:1602.05924] between our recent result [T.S. ... Our result, proven in [

*Cubitt*et al. 2015a,b] , is: Result 3 (Our result). Problem 2 is undecidable. (This also immediately implies that Problem 1 is undecidable.) ...

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An Information-Theoretic Proof of the Constructive Commutative Quantum Lovász Local Lemma
[article]

2013
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arXiv
*
pre-print

*S*.

*Cubitt*. Preparing topological projected entangled pair states on a quantum computer. Physical Review A, 88(032321), 2013. [VWC09] F. Verstraete, M. M. Wolf, and J. I. Cirac. ...

*s*].nbr ←

*S*[

*s*].nbr + 1 mod k 42: if

*S*[

*s*].nbr = 0 then 43:F [l] ← F [l] + 1 if F [l] then 47:

*s*←

*s*− 1

*S*[

*s*+ 1].proj ←

*S*[

*s*+ 1].proj − Γ + (

*S*[

*s*].proj,

*S*[

*s*].nbr) term[l + 1] ← term[l + 1] + 1 log(N ) log ...

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A constructive commutative quantum Lovasz Local Lemma, and beyond
[article]

2013
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arXiv
*
pre-print

. , Π at ) = G] = s:G(

arXiv:1112.1413v2
fatcat:i5a5hb7eivho7cublvucquvvam
*s*)=G Pr(*s*) ≤*s*∈*S*p G (*s*) · t i=1 Tr[Π a i ρ] = v∈G Tr[Π v ρ] ·*s*∈*S*p G (*s*) = v∈G Tr[Π v ρ]. (52) Partial Resample DAGs Later on, we will need to consider a variant of the resample ...*Cubitt*et al. [2009] ). Either way, this means that the problem instance itself sets a lower-bound on the spectral gap. ...##
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Undecidability of the Spectral Gap in One Dimension

2020
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Physical Review X
*

(Phase-estimation QTM,

doi:10.1103/physrevx.10.031038
fatcat:7jfxyubuhfasbb5p7fduzat5py
*Cubitt*et al. ... This phase estimation QTM is the only ingredient we require from*Cubitt*et al. ...##
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The Complexity of Relating Quantum Channels to Master Equations
[article]

2011
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arXiv
*
pre-print

Mathematically, such a Markovian master equation generates a one-parameter (time t) semi-group (evolving for time t and then time

arXiv:0908.2128v2
fatcat:7g7boizeirag3j6fcd2k5obilq
*s*is equivalent to evolving for time t +*s*) of CPT maps (the evolution ... If we choose the first n C diagonal elements of*S*to be 1/2, and choose the next n v diagonal elements of*S*to be 5/6, then it is straightforward to verify that the equations in the ccp condition of Eq ...##
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Complexity Classification of Local Hamiltonian Problems

2014
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2014 IEEE 55th Annual Symposium on Foundations of Computer Science
*

One way of making the problem more physically meaningful is to restrict the Hamiltonian in question by picking its terms from a fixed set

doi:10.1109/focs.2014.21
dblp:conf/focs/CubittM14
fatcat:j6iwgcvi3zchpai3kukmcohpny
*S*. ... whose terms are picked from a fixed set*S*correspond to problems either in P; polynomial-time equivalent to the Ising model with transverse magnetic fields; or QMAcomplete. ... and 1'*s*. ...##
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An Extreme form of Superactivation for Quantum Zero-Error Capacities
[article]

2011
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arXiv
*
pre-print

*S*.

*Cubitt*was supported by a Leverhulme early-career fellowship, and through the integrated EC project "QAP" (contract no. IST-2005-15848). G. ...

*S*.

*Cubitt*is with the Departamento de Análisis Matemático, Universidad Complutense de Madrid, Plaza de Ciencias 3, Ciudad Universitaria, 28040 Madrid, Spain (email: tcubitt@mat.ucm.es). G. ...

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Entanglement can Completely Defeat Quantum Noise

2011
*
Physical Review Letters
*

*S*. is supported by the DARPA QuEST contract HR0011-09-C-0047. Tr½E 1 E 2 ðc ÞE 1 E 2 ð'Þ ¼ 0 [Eq. ... i, where X is the unitary consisting of 1'

*s*down the anti-main diagonal (i.e., one possible generalization of the Pauli matrix x ). ...

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