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History-state Hamiltonians are critical [article]

Carlos E. González-Guillén, Toby S. Cubitt
2018 arXiv   pre-print
Pr U ∼νs |f h,W (U )| ≥ δ 2 ≤ 8 6144s d n δ 2 s/2 .  ...  ) that Pr U ∼νs |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 .  ... 
arXiv:1810.06528v1 fatcat:kxfdeg7myjgh7brkd6pc2iwmvm

Uncomputably Complex Renormalisation Group Flows [article]

James D. Watson, Emilio Onorati, Toby S. Cubitt
2021 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 (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  ... 
arXiv:2102.05145v1 fatcat:pemuiuogxzgv3oatda3oel4dku

Uncomputability of Phase Diagrams [article]

Johannes Bausch, Toby S. Cubitt, James D. Watson
2019 arXiv   pre-print
This work builds off recent results from 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.  ... 
arXiv:1910.01631v1 fatcat:azmuyephnbhy3gy7sxttcurn4q

Entanglement in the stabilizer formalism [article]

David Fattal, Toby S. Cubitt, Yoshihisa Yamamoto, Sergey Bravyi, Isaac L. Chuang
2004 arXiv   pre-print
We further define the local subgroup 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 |.  ... 
arXiv:quant-ph/0406168v1 fatcat:pjxw4yea2zaajdz6lpsnvqizr4

Universal Refocusing of Systematic Quantum Noise [article]

Imdad S. B. Sardharwalla, Toby S. Cubitt, Aram W. Harrow, Noah Linden
2016 arXiv   pre-print
holds for all 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  ... 
arXiv:1602.07963v1 fatcat:bhk6wy4m2jhefd57ndj6ivifhm

Computational Complexity of the Ground State Energy Density Problem [article]

James D. Watson, Toby S. Cubitt
2021 arXiv   pre-print
Lockhart, 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.  ... 
arXiv:2107.05060v1 fatcat:5nvhdl6f2ndsvorirbss75t2vy

Are problems in Quantum Information Theory (un)decidable? [article]

Michael M. Wolf, Toby S. Cubitt, David Perez-Garcia
2011 arXiv   pre-print
Given a Boolean combi- nation φ(X, A) of the F i '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).  ... 
arXiv:1111.5425v1 fatcat:tivo3fmt3new3neekoe43fs76y

Comment on "On the uncomputability of the spectral gap" [article]

Toby S. Cubitt, David Perez-Garcia, Michael M. Wolf
2016 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.)  ... 
arXiv:1603.00825v1 fatcat:cib6crzgljcbdpq22lnqnrogfq

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

Martin Schwarz, Toby S. Cubitt, Frank Verstraete
2013 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: ss − 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  ... 
arXiv:1311.6474v1 fatcat:dafl3h2mrvffrn27few3pz3gae

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

Toby S. Cubitt, Martin Schwarz
2013 arXiv   pre-print
. , Π at ) = G] = s:G(s)=G Pr(s) ≤ sS p G (s) · t i=1 Tr[Π a i ρ] = v∈G Tr[Π v ρ] · sS 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.  ... 
arXiv:1112.1413v2 fatcat:i5a5hb7eivho7cublvucquvvam

Undecidability of the Spectral Gap in One Dimension

Johannes Bausch, Toby S. Cubitt, Angelo Lucia, David Perez-Garcia
2020 Physical Review X  
(Phase-estimation QTM, Cubitt et al.  ...  This phase estimation QTM is the only ingredient we require from Cubitt et al.  ... 
doi:10.1103/physrevx.10.031038 fatcat:7jfxyubuhfasbb5p7fduzat5py

The Complexity of Relating Quantum Channels to Master Equations [article]

Toby S. Cubitt, Jens Eisert, Michael M. Wolf
2011 arXiv   pre-print
Mathematically, such a Markovian master equation generates a one-parameter (time t) semi-group (evolving for time t and then time 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  ... 
arXiv:0908.2128v2 fatcat:7g7boizeirag3j6fcd2k5obilq

Complexity Classification of Local Hamiltonian Problems

Toby S. Cubitt, Ashley Montanaro
2014 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 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.  ... 
doi:10.1109/focs.2014.21 dblp:conf/focs/CubittM14 fatcat:j6iwgcvi3zchpai3kukmcohpny

An Extreme form of Superactivation for Quantum Zero-Error Capacities [article]

Toby S. Cubitt, Graeme Smith
2011 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.  ... 
arXiv:0912.2737v2 fatcat:mxmnpowuhbbi7enrgmxkzfubsu

Entanglement can Completely Defeat Quantum Noise

Jianxin Chen, Toby S. Cubitt, Aram W. Harrow, Graeme Smith
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 ).  ... 
doi:10.1103/physrevlett.107.250504 pmid:22243059 fatcat:5zvit5wdqjbpxbmrzsxvu4duw4
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