Quantum Error Correction in a Correlated Environmentprofs.if.uff.br/paraty07/paraty09/palestras_ws/Eduardo_Novais.pdf · Basic Assumptions We want QEC to succeed, hence we make some

Quantum Error Correction in a Correlated Environment

Eduardo Novais1,3, E. R. Mucciolo2 and Harold U. Baranger1

1- Department of Physics, Duke University, Durham NC 27708-0305.

2- Department of Physics, University of Central Florida, Orlando FL 32816-2385.

3- CCNH, Universidade Federal do ABC, Santo André SP Brazil.

September 2009.

Eduardo Novais (UFABC) Quantum Error Correction in a Correlated Environment September 2009. 1 / 42

Quantum Computation:“when quantum mechanics meets computer science’.


Quantum Error Correction

Usual assumptions in the traditional QEC theoryfast gates;

fast measurements;

error models


Quantum Error Correction

“threshold theorem”Provided the noise strength is below a critical value, quantum informationcan be protected for arbitrarily long times.

Hence, the computation is said to be fault tolerant or resilient.

Dorit Aharonov, Phys. Rev. A 062311 (2000).Quantum to classical phase transition in a noisy QC.

(ε)error probability of a qubit

noise quantum computerresilient quantum computation

threshold theorem

strong entanglement between qubits

’’low temperature regime"

simulated efficiently by a Turing machine

weak entanglement between qubits

’’high temperature regime"


What if we would like to start from a microscopic model?

The standard prescription:

quantum master equation and

dynamical semi-groups.

This is a natural approach:The computer is the object of interest; hence one starts the discussion byintegrating out the environmental degrees of freedom.

The price that we usually pay:

1 First, Born approximation;2 then, the Markov approximation.

QEC is a perturbative method!






































What is the problem with the B-M approximation?

R. Alicki, Daniel A. Lidar and Paolo Zanardi, PRA 73 052311 (2006).Internal Consistency of Fault-Tolerant Quantum Error Correction inLight of Rigorous Derivations of the Quantum Markovian Limit.... These assumptions are: fast gates, a constant supply of fresh cold ancillas,and a Markovian bath. We point out that these assumptions may not bemutually consistent in light of rigorous formulations of the Markovianapproximation. ...


What is the problem with the B-M approximation?

Ulrich Weissin Quantum Dissipative SystemsWhile the Markov assumption can easily be dropped, the more severelimitation of this method is the Born approximation for the kernel. Inconclusion, the Born-Markov quantum master equation method provides areasonable description in many cases, such as in NMR, in laser physics, andin a variety of chemical reactions. However, this method turned out to be notuseful in most problems of solid state physics at low temperatures for whichneither the Born approximation is valid nor the Markov assumption holds.


Correlated environments

We do not want to assume Born-Markov approximation.

Some references that also do not assume B-M:Knill, Laflamme, Viola, PRL 84, 2525 (2000),

Terhal and Burkard, PRA 71, 012336,

Aliferis, Gottesman, Preskill, QIC 6, 97 (2006),

Aharonov, Kitaev, Preskill, PRL 96, 050504 (2006),

etc...


How correlated environments have been treated before?

define the evolution with at least one error

E (t) = U (t)−1 =− ih

Z t

0dt′V

(t′)

U(t′)

.

find a bound for the norm of E

||E (t)|| ≤ 1h

Z t

0dt′∣∣∣∣V (t′)∣∣∣∣≤ Λt

h,

where Λ as the largest eigenvalue of V .

The problemit only makes sense when: ||E || � 1, thus

how to deal with the case |Λt| � 1?


Perturbation theoryfree Hamiltonian: H0;a “perturbation”:

V(t) = λ

Z L

0dx f (x, t) ,

“two point correlation function:

〈Ψ |f (x1, t1) f (x2, t2)|Ψ〉 ∼ F

(1

(∆x)2δ,

1

(∆t)2δ/z

),

.

〈Ψ|E† (t)E (t) |Ψ〉2

= 1−〈Ψ|Tt cos[

1h

Z t

0dt′V(t′)

]|Ψ〉 .

“RPA” series gives ∼λ2m(Lt)2m(D+z−δ).Perturbation theory around the “non-interacting” ground state is a badstart for D+ z−δ > 0.


A schematic phase-diagram

How QEC changes this picture?


A useless summary of QEC

Steps in a QEC evolution1 encode the information in a

large Hilbert space;2 let the system evolves;3 extract the “syndrome”;4 correct the system;5 start again;





t0 ∆ 2∆

}Error Correction CycleSyndrome Extraction






Quantities to calculateWhen a probability can be defined:

probability of a history of syndromes

p0 probability of nothaving an error;

p1 probability ofhaving an error.

p0

p1

p0

p1

p0

p1

p1

p1

p0

p0

t

residual decoherenceIf error correction is not perfect (and usually it is not), there is always someresidual decoherence to evaluate.


How this translates to the quantum evolution?

consider and environment controlled by a “free” Hamiltonian:

H0

and an interaction:

V = ∑x,α

λαfα (x)σα (x) , (1)

where~f is a function of the environment degrees of freedom and~σ are thePauli matrices that parametrize the qubits.


Time evolution of encoded qubits

usually the quantum evolution in the interaction picture is:

U (∆,0) = Tte−i λ

2 ∑xR

∆

0 dt~f (x,t)~σ(x)

= 1− iλ

2 ∑x

Z∆

0dt~f (x, t)~σ(x)

− λ2

4 ∑x,y

α,β = {x,y,z}

Z∆

0dt1

Z t1

0dt2f α (x, t1) f β (y, t2)σ

α (x)σβ (y)+ ...

when the syndrome is extracted only a set of terms is kept to the nextQEC cycle, in other words,

each syndromes defines a different “evolution” operator for amacrolocation.


Calculating the probability of an evolution

We start with:

ψ

ψ

A

A B

B

We would like to write:

ψA B

ψA B

t


How to deal with the general problem?

Executive summarystart with a Hamiltonian,

define a local error probability,

identify the long range operator,

study how long range component altersthe local part.


























Returning to the general problemFor an environment controlled by a “free” Hamiltonian:

H0

there are three important parameters:1 the number of spatial dimensions, D;2 the wave velocity, v and;3 dynamical exponent, z.

a general form for the interaction is:

V = ∑x,α


where~f is a function of the environment degrees of freedom and~σ arethe Pauli matrices that parametrize the qubits.Hence, the evolution operator is

U (∆,λα) = Tt e−iR

∆

0 dt ∑x,α λαfα(x,t)σα(x) , (3)



H0



V = ∑x,α




∆




H0



V = ∑x,α




∆




H0



V = ∑x,α




∆




H0



V = ∑x,α




∆




H0



V = ∑x,α




∆




H0



V = ∑x,α




∆



Defining a coarse-grain grid

Hypercube

of size ∆ in time and ξ = (v∆)1z in space.

ξ

∆



Hypercube


HypothesisThere is only one qubit ineach hypercube.

allows to define theprobability of anerror in a qubit.

for “short times” itis an impurityproblem. ξ

∆



Hypercube





∆



Hypercube





∆



Hypercube





∆



Hypercube





∆



Hypercube





∆



Hypercube





∆



2∆

∆

2ξξ 3ξ x

t



2∆

∆

2ξξ 3ξ x

t



2∆

∆

2ξξ 3ξ x

t


Qubit evolution in a QEC cycleLowest order perturbation theory for an error of type α.

υα (x1,λα)≈−iλα

Z∆

0dt fα (x1, t) ,

Note: QEC removed the qubit operator!!! I only need to use the field’scommutation relations.

Perturbation theory improved by RG

υα (x1,λα) ≈ −iλα

Z∆

0dt fα (x1, t)

− 12

∣∣εαβγ

∣∣λβλγ σα (∆)Tt

Z∆

0dt1 dt2 fβ (x1, t1) fγ (x1, t2)σβ (t1)σγ (t2)

+i6 ∑

β

λαλ2β

σα (∆)

× Tt

Z∆

0dt1 dt2 dt3 fα (x1, t1) fβ (x1, t2) fβ (x1, t3)σα (t1)σβ (t2)σβ (t3) ,

ξ

∆




Z∆

0dt fα (x1, t) ,



dλα

d`= gβγ (`)λβλγ +∑

β

hαβ (`)λαλ2β,

integrate from the ultraviolet cut-off to ∆−1, defines λ∗.

ξ

∆




Z∆

0dt fα (x1, t) ,



υα (x1,λ∗α)≈−iλ∗α

Z∆

0dt fα (x1, t) ,

ξ

∆


Separating intra- and inter-hypercube components

Calculating a probability

P(...;α,x1; ...)≈⟨...υ†

α (x1,λ∗α) ...υα (x1,λ

∗α) ...

⟩.

υ2α (x1,λ

∗α)≈ εα + (λ∗α∆)2 : |fα (x1,0)|2 : ,

εα = (λ∗α)2Z

∆

0dt1

Z∆

0dt2⟨f †α (x1, t1) fα (x1, t2)

⟩




P(...;α,x1; ...)≈⟨...υ†

α (x1,λ∗α) ...υα (x1,λ

∗α) ...

⟩.

υ2α (x1,λ

∗α)≈ εα + (λ∗α∆)2 : |fα (x1,0)|2 : ,

εα = (λ∗α)2Z

∆

0dt1

Z∆

0dt2⟨f †α (x1, t1) fα (x1, t2)

⟩

t0 ∆ 2∆

Error Correction Cycle




P(...;α,x1; ...)≈⟨...υ†

α (x1,λ∗α) ...υα (x1,λ

∗α) ...

⟩.

υ2α (x1,λ

∗α)≈ εα + (λ∗α∆)2 : |fα (x1,0)|2 : ,

εα = (λ∗α)2Z

∆

0dt1

Z∆

0dt2⟨f †α (x1, t1) fα (x1, t2)

⟩

t∆ 2∆0

Error Correction Cycle


Probability of having m errors after N steps in R qubits

Pαm = pm

Z dx1

(v∆)D/z ...dxm

(v∆)D/z

Z N∆

0

dt1∆

...Z tm−1

0

dtm∆

×

⟨[∏

ζ

F0(xζ, tζ)][

1+Fα(x1, t1)]...[1+Fα(xm, tm)

]⟩

F0 (x1,0) = 1−∑β

(λ∗

β∆

)2:∣∣fβ (x1,0)

∣∣2 :

1−∑β=x,y,z εβ

,

Fα (x1,0) =(λ∗α∆)2

εα

: |fα (x1,0)|2 : .

“New” perturbation theoryin the coarse grain grid.

t

ξEduardo Novais (UFABC) Quantum Error Correction in a Correlated Environment September 2009. 23 / 42


zeroth order terms is just the stochastic probability:

t

ξ

pm

Z m

∏k=1

dxk

(v∆)D/z

dtk∆

= pm

(NRm

).



second order term is the first correction due to long range correlations

t

ξ

pm

Z m

∏k=1

dxk

(v∆)D/z

dtk∆〈Fα (xi, ti)Fα (xj, tj)〉 .



correlation function

〈Fα (xi, ti)Fα (xj, tj)〉 ∼ F(|xi−xj|−4δα , |ti− tj|−4δα/z

)where δα is the scaling dimension of fα

using scaling again

dλ∗αd`

= (D+ z−2δα)λ∗α.

defines the stability of the perturbation theory:1 D+ z−2δα < 0 corrections are small,2 D+ z−2δα > 0 new derivation needed.






using scaling again

dλ∗αd`

= (D+ z−2δα)λ∗α.







using scaling again

dλ∗αd`

= (D+ z−2δα)λ∗α.



Threshold theorem as a quantum phase transition

low temperature -disentangled phase

|ψtotal〉= |ψcomputer〉⊗ |ψenvironment〉

1 qubits and environment areweakly entangled,

2 the qubits are stronglyentangle among themselves,

3 low decoherence.

high temperature -entangled phase

1 qubits and environment arestrongly entangled,

2 the qubits are weakly entangleamong themselves,

3 strong decoherence.

low temperature -entangled phase

1 computer and qubits arestrongly entangle,

2 strong decoherence.


Schematic phase diagram


Schematic phase diagram


Gaussian noise - More of the same ?

Fault-tolerant quantum computation versus Gaussian noiseby Hui Khoon Ng, John PreskillPhys. Rev. A 79, 032318 (2009)get to the same criteria for the threshold theorem (d + z−2δ), but does not

need the hypercubes ( )

We do not agree with this conclusion (hopefully John will invite me back

to Caltech ).

All our assumptions will be favorable to QEC, and therefore our resultsshould be regard as a lower bound to the effects of quantum correlations.


the model

We choose the simplest case: We assume that each component of a qubitis couple with a free bosonic bath.

HI = ∑x

∑α={x,y,z}

λα : f α (x) : σαx

: f α (x) : = ∑k6=0

gα,|k|eik.x

a†α,k +g∗

α,|k|e−ik.x

aα,k

where g is a function of k and aα,k obeys standard commutation relations.

If we had choose a more elaborated algebraic structure (like a Kacalgebra), lower frequency excitations in one direction produce a showerof excitations in the others.

We also use the :: to denote normal ordering.


Basic Assumptions

We want QEC to succeed, hence we make some assumptions that arefavorable:

1 We will consider fast gates that are done perfectly,2 we will disregard all high frequency component of the noise by keeping

the lowest order terms on the quantum evolution on each QEC step,3 we will assume the all measurements are done perfectly,4 we will assume that state preparation is also done perfectly,5 we will consider the most favorable quantum computer evolution, where

ALL the syndromes returned a result of no-error (all other evolutionswill give us a worst result).

The single assumption that we take that is not favorable to QEC is:1 All qubits of a logical qubit are inside a hypercube.


The QEC Code

For simplicity we will consider the smallest distance 1 code (the 5 qubitcode).

The 5-qubit code is defined by the stabilizers and logical operations

g1 = xzzx1

g2 = 1xzzx

g3 = x1xzz

g4 = zx1xz

X = xxxxx

Y = yyyyy

Z = zzzzz

All these operators keep thelogical kets into the logicalHilbert space

It is straightforward to extendour calculation to ANYstabilizer code.


The QEC CodeL =

{1, X, Y, Z

}is a normal subgroup.

the stabilizers generate Abelian group, where each element is its owninverse.

S ={

1,gi,gigj>i,gigj>igk>j, ....,g1g2...gn}

.

For the particular case of the 5-qubit code we define the notationelement definition

1 11111g1 xzzx1g2 1xzzxg3 x1xzzg4 zx1xzh1 xy1yxh2 1zyyzh3 yyz1zh4 xxy1y

element definitionh5 z1zyyh6 yxxy1h7 1yxxyh8 zyyz1h9 y1yxxh10 yz1zyh11 zzx1x


The QEC codeWe will consider only the syndrome that correspond to no-error.

1 X Y Z1 11111 xxxxx yyyyy zzzzzg1 xzzx1 1yy1x zxxzy y11yzg2 1xzzx x1yy1 yzxxz zy11yg3 x1xzz 1x1yy zyzxx yzy11g4 zx1xz y1x1y xzyzx 1yzy1h1 xy1yx 1zxz1 z1y1z yxzxyh2 1zyyz xyzzy yx11x z1xx1h3 yyz1z zzyxy 11xyx xx1z1h4 xxy1y 11zxz zz1y1 yyxzxh5 z1zyy yxyzz xyx11 1z1xxh6 yxxy1 z11zx 1zz1y xyyxzh7 1yxxy xz11z y1zz1 zxyyxh8 zyyz1 yzzyx x11xy 1xx1zh9 y1yxx zxz11 1y1zz xzxyyh10 yz1zy zyxyz 1xyx1 x1z1xh11 zzx1x yy1x1 xxzyz 11yzy


Quantum Evolution in a QEC Period

U (∆,0) = Tte−iR

∆

0 dtHI(t)

We are assuming that lowest order perturbation theory is valid on eachqubit.

v0 (∆,0) = 1+ ∏α6=β

(λα)2(

λβ

)2 Z∆

0dt1dt2dt3dt4

[5

∑i=1

∑j6=k 6=l 6=i

: f α (i, t1) :: f α (j, t2) :: f β (k, t3) :: f β (l, t4) :

]1

+ iλx

Z∆

0dt1dt2dt3

[∑

α=y,z(λα)2

5

∑i=1

∑j 6=k 6=i

: f x (i, t1) :: f α (j, t2) :: f α (k, t3) :

]X

+ iλy

Z∆

0dt1dt2dt3

[∑

α=x,z(λα)2

5

∑i=1

∑j 6=k 6=i

: f y (i, t1) :: f α (j, t2) :: f α (k, t3) :

]Y

+ iλz

Z∆

0dt1dt2dt3

[∑

α=x,y(λα)2

5

∑i=1

∑j 6=k 6=i

: f z (i, t1) :: f α (j, t2) :: f α (k, t3) :

]Z

− iλx(λy)2 (λz)2

Z∆

0dt1dt2dt3dt4dt5

[5

∑i=1

∑j 6=k 6=l 6=m 6=i

: f x (i, t1) :: f y (j, t2) :: f y (k, t3) :: f z (l, t4) :: f z (m, t5) :

]X

− iλy (λx)2 (λz)2Z

∆

0dt1dt2dt3dt4dt5

[5

∑i=1

∑j 6=k 6=l 6=m 6=i

: f y (i, t1) :: f x (j, t2) :: f x (k, t3) :: f z (l, t4) :: f z (m, t5) :

]Y

− iλz(λy)2 (λx)2

Z∆

0dt1dt2dt3dt4dt5

[5

∑i=1

∑j 6=k 6=l 6=m 6=i

: f z (i, t1) :: f y (j, t2) :: f y (k, t3) :: f x (l, t4) :: f x (m, t5) :

]Z


Quantum Evolution in a QEC PeriodNow we re-write the quantum evolution after normal ordering

v0 (∆,0) = 1−5iλx

[−2(λ∗y)2−2

(λ∗z)2 +

(λ∗y)2 (

λ∗z)2]Z

∆

0dt

5

∑i=1

f x (xi, t) X

+ 5iλy

[−2(λ∗x)

2−2(λ∗z)2 +(λ∗x)

2 (λ∗z)2]Z

∆

0dt

5

∑i=1

f y (xi, t) Y

+ 5iλz

[−2(λ∗y)2−2(λ∗x)

2 +(λ∗y)2 (λ∗x)

2]Z

∆

0dt

5

∑i=1

f z (xi, t) Z

+ normal order terms

Note now that if are interested in things that happen between hypercubes, thenwe can use

Z∆

0dt

5

∑i=1

f α (xi, t)≈ 5∆f α (x,0)

where x is the coarse grain position variable for the logical qubit.Eduardo Novais (UFABC) Quantum Error Correction in a Correlated Environment September 2009. 36 / 42

Quantum Evolution with no-errorsAfter integrating the high frequency part, the coupling isrenormalized(downwards or upwards is model dependent).The leading terms are

v0 (∆,0) = 1− i∆λxf x (x,0) X

− i∆λyf y (x,0) Y

− i∆λzf z (x,0) Z

We can re-exponentiate and write the Dyson series for the evolution operatorof a logical qubit in a QEC evolution where only no-error syndromes werefound.

U0 = Tte−iR T

0 dtλxf x(x,t)X+λyf y(x,t)Y+λzf z(x,t)Z

where dt = ∆ and T = N∆ with N the total number of QEC steps performed.This is the low frequency theory of a perfect evolution under QEC.


Quantum Evolution with no-errors

The new Dyson Seriesis IDENTICAL to the high frequency theory, where we just replaced thephysical qubit by the logical one and the parameters were renormalizedsomewhat downwards/upwards

U0 = Tte−iR T




The new Dyson Seriesis IDENTICAL to the high frequency theory, where we just replaced thephysical qubit by the logical one and the parameters were renormalizedsomewhat downwards/upwards

U0 = Tte−iR T




U0 = Tte−iR T


Important ConclusionsThe relevant scaling dimension for this problem is δ, not 2δ as stated on Phys.Rev. A 79, 032318 (2009).For any quantum QEC protocol, where at least one logical qubit is containedin a hypercube, we can define a time scale where uncorrectable errors willdestroy the computation.

λ(t = N∆) = λ [N]D+z−δ



Although QEC can sometimes improve the time scales, resilience cannot beachieved if

1 D+ z−2∆, for systems with one physical qubit per hypercube.2 D+ z−∆, for systems with at least one logical qubit per hypercube.


Conclusions

We...1 ... studied decoherence in quantum computers in a correlated

environment,2 ... used an Hamiltonian description,3 ... derived when is possible to use the “threshold theorem”,4 ... produced a parallel with the theory of quantum phase transitions.5 ... discussed the entropy production due to uncorrectable errors.

part of this work can be found in1 Phys. Rev. Lett. 97, 040501 (2006).

2 Phys. Rev. Lett. 98, 040501 (2007).

3 Phys. Rev. A. 78: art. 012314 (2008).

4 in the near future as a chapter in a book edited by Daniel Lidar, Paolo Zanardi and ToddBrun (published by Cambridge Press).


Documents

Quantum Error Correction in a Correlated Environmentprofs.if.uff.br/paraty07/paraty09/palestras_ws/Eduardo_Novais.pdf · Basic Assumptions We want QEC to succeed, hence we make some