zeno effect for quantum computation and...

IntroductionResults

Conclusions

Zeno effect forquantum computation and control

Gerardo Paz-Silva(1), Ali Rezakhani(1,2), Jason Dominy(1),Daniel Lidar(1)

(1)Center for Quantum Information Science & Technology,University of Southern California

(2)Sharif University of Technology, Tehran

2nd International Conference on Quantum Error CorrectionUniversity of Southern California

December 7, 2011

Gerardo Paz-Silva, Ali Rezakhani, JMD, Daniel Lidar arXiv:1104.5507

IntroductionResults

Conclusions

The problemThe solutions

That unavoidable bath

System: e.g., n qubits

coupled to

Bath: uncontrollable, unknown, and devious, butbounded in operator norm

H = HS ⊗HB

H = HS ⊗ IB + IS ⊗ HB + HSB

USB = eiHT

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coupled to

H = HS ⊗HB

USB = eiHT

Decoherence ...

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coupled to

H = HS ⊗HB

USB = eiHT

Decoherence ...

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From the arsenal of anti-decoherence solutions

Passive methods:decoherence-free subspaces

Active methods:Closed-loop

Quantum error correction (QEC)Open-loop

Dynamical decouplingQuantum Zeno (QZ) effect: (Misra & Sudarshan, 1977)well known method for protection of given states usingprojective measurements

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This talk

DesiderataCan we protect an arbitrary state or subspace usingthe QZE?If so, how well, given finite resources? I.e.,finite number of non-projective (weak) measurements

IntroductionResults

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This talk

DesiderataCan we protect an arbitrary state or subspace usingthe QZE?If so, how well, given finite resources? I.e.,finite number of non-projective (weak) measurements

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QZ cartoon

measurements slow down system evolution; “watched pot never boils”

M weak measurements in time T , both M,T <∞

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QZ cartoon

measurements slow down system evolution; “watched pot never boils”

Mao, undated

M weak measurements in time T , both M,T <∞

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Weak measurement cartoon

HS-probe

Probes interact with the measured system for a finite time δ viaa weak coupling HS−probe.

Measurement strength ε ∝ δ‖HS−probe‖.

Probes are measured projectively, but S is measured “weakly".

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Measure & compute

How can we use weak-measurement QZ to protectarbitrary states and simultaneously quantumcompute/control?

Main idea:Encode into a stabilizer quantum error correction (ordetection) codeWeakly measure stabilizer elements or generators of theQECCompute/control by applying the logical operators(elements of normalizer) of the QEC as Hamiltonians.They commute with measurements!

IntroductionResults

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Measure & compute

How can we use weak-measurement QZ to protectarbitrary states and simultaneously quantumcompute/control?

Main idea:Encode into a stabilizer quantum error correction (ordetection) codeWeakly measure stabilizer elements or generators of theQECCompute/control by applying the logical operators(elements of normalizer) of the QEC as Hamiltonians.They commute with measurements!

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Youkzee – (Encoded Weak Quantum Zeno Effect) protocolWeak measurements results

The weak stabilizer group measurement protocol

Task: Protect an unknown state, in a known code, for time T .Given:

H = HS + HB + HSB; J0 ≡ 2‖HS + HB‖ and J1 ≡ 2‖HSB‖Free evolution superoperator U(t) = e−it[H,·]

HSB at most d − 1 local: HSB =∑d−1

k=1 HSB(k)

Encode into [[n, k ,d ]] code with stabilizer group G(we’re given that HSB doesn’t contain any terms that act aslogical operators)

Apply weak measurements P̂(ε,G) of G’s elements everytime interval T/M:

weak stabilizer-group measurement protocol

%S(T ) = TrB

[(P̂(ε,G)U(T/M)

)M%SB(0)

]Gerardo Paz-Silva, Ali Rezakhani, JMD, Daniel Lidar arXiv:1104.5507

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k=1 HSB(k)

%S(T ) = TrB

[(P̂(ε,G)U(T/M)

)M%SB(0)

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k=1 HSB(k)

%S(T ) = TrB

[(P̂(ε,G)U(T/M)

)M%SB(0)

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k=1 HSB(k)

%S(T ) = TrB

[(P̂(ε,G)U(T/M)

)M%SB(0)

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Protection with weak measurements

Want the protocol to yield dynamics close to the casewhere HSB = 0

D[%S(T ), %0S(T )] = ‖%S(T )− %0

S(T )‖

%S(T ) is the state under evolution with Zeno protocol%0

S(T ) is the state under free evolution with HSB = 0.

Decompose this distanceDyson seriesStabilizer code algebraic structure – error subspacesTriangle inequalities

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Want the protocol to yield dynamics close to the casewhere HSB = 0

D[%S(T ), %0S(T )] = ‖%S(T )− %0

S(T )‖

%S(T ) is the state under evolution with Zeno protocol%0

S(T ) is the state under free evolution with HSB = 0.

Decompose this distanceDyson seriesStabilizer code algebraic structure – error subspacesTriangle inequalities

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Figure: Sister Celine (Mary Celine Fasenmyer). “The subject ofcomputerized proofs of identities begins with the Ph.D. thesis ofSister Celine at the University of Michigan in 1945.”† Her thesisshowed how one can find recurrence relations that are satisfied bysums of hypergeometric terms in a purely algorithmic way.

†M. Petkovšek, H. S. Wilf, and D. Zeilberger, A=B (1996)Gerardo Paz-Silva, Ali Rezakhani, JMD, Daniel Lidar arXiv:1104.5507

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Sufficiently many finite-strength measurementsapproximate the ideal limit arbitrarily well.

D[%S(T ), %0S(T )] ≤

(1− h(J0 − J1)

Γg(M)

Γ1(M)

)[1 + Γ1(M)

]M− h(J1 − J0)[1 + Γg(M)

]M+ Γg(M)

∑s=+,−

As(M)γM−1s (M)− eTJ0=: B

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and the definitions...

h(x) :=

{0 x < 01 x ≥ 0

(Heaviside function)

β(M) :=

Γ1(M) := 1Q+1 e

QTJ1M + Qe−

)− 1 J0 ≥ J1

Γg(M) := 1Q+1 e

QTJ1M − e−

)J0 ≤ J1

γ± :=12(1 + β + (1 + Qβ)ζq)

√(1 + β − (1 + Qβ)ζq

)2+ 4Qβ2ζq

A± :=Qβζq(γ± + β) + (1 + β)

[(1 + β)− γ∓

]γ± − γ∓

ζ := sech(ε)

q := (Q + 1)/2 ←− every error anticommutes with exactly1/2 of all stabilizer elements

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Asymptotics of the weak measurements bound

Can prove that for any measurement strength ε > 0:

B =QTJ1

2+ eTJm

(1+TJm

) sechq(ε)

1−sechq(ε)

)M→∞−→ 0

whereM is the number of measurements (i.e. iterations of theprotocol).q := (Q + 1)/2 = |S|/2 is half the order of the stabilizergroupJm := max{J0, J1}

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Measuring generators vs full stabilizer group

Cheaper option: measure only stabilizer generators.Exponentially more efficient in the number ofsimultaneous binary measurements but slowerconvergence of bound to zero.We use the fact that each error anticommutes with at leastone generator⇒ Same bounds but with q = 1.

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Measuring generators vs full stabilizer group

Cheaper option: measure only stabilizer generators.Exponentially more efficient in the number ofsimultaneous binary measurements but slowerconvergence of bound to zero.We use the fact that each error anticommutes with at leastone generator⇒ Same bounds but with q = 1.

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Bound as a function of M and εJ0T = 1, Q̄ = 4, J1/J0 = 1/10

log10M

log10B

Figure: Bottom: strong limit (B∞). Middle: weak, full (B|q=2Q̄−1 ). Top: weak, generators (B|q=1)

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Summary and future research

Found rigorous distance bound on weak stabilizermeasurement protocol for protection of arbitrary states orsubspacesShown that in the limit of large number of measurementscan protect encoded information arbitrarily wellWorks for time-dependent Hamiltonians⇒ applications toAQCArgued that we can “compute while protecting” usinglogical operatorsWhat about selective measurements? Do the individualoutcomes exhibit a ‘concentration of measure’ about theaverage as occurs with projective measurements?

IntroductionResults

Conclusions

Summary and future research

Found rigorous distance bound on weak stabilizermeasurement protocol for protection of arbitrary states orsubspacesShown that in the limit of large number of measurementscan protect encoded information arbitrarily wellWorks for time-dependent Hamiltonians⇒ applications toAQCArgued that we can “compute while protecting” usinglogical operatorsWhat about selective measurements? Do the individualoutcomes exhibit a ‘concentration of measure’ about theaverage as occurs with projective measurements?

zeno effect for quantum computation and...

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