5-qubit Quantum error correction in a charge qubit quantum computer

Dave Touchette, Haleemur Ali and Michael HilkeDepartment of Physics, McGill University, Montreal, Quebec, H3A 2T8, Canada

(Dated: October 18, 2010)

We implement the DiVincenzo-Shor 5 qubit quantum error correcting code into a solid-state quan-tum register. The quantum register is a multi charge-qubit system in a semiconductor environment,where the main sources of noise are phase decoherence and relaxation. We evaluate the decay ofthe density matrix for this multi-qubit system and perform regular quantum error corrections. Theperformance of the error correction in this realistic system is found to yield an improvement ofthe fidelity. The fidelity can be maintained arbitrarily close to one by sufficiently increasing thefrequency of error correction. This opens the door for arbitrarily long quantum computations.

PACS numbers: 03.67.Pp, 03.67.Lx, 85.35.BeKeywords: quantum error correction, 5-qubit code, solid-state quantum register, multi charge-qubit system,phase decoherence, relaxation

I. INTRODUCTION

The task of building a quantum computer, allowingfor massive quantum parallelism [1], long-lived quantumstate superposition and entanglement, is one of the bigchallenge of twenty-first century physics. It requires pre-cise maintenance and manipulation of quantum systemsat a level far from anything that has ever been done be-fore. Many criteria, nicely summarized by the DiVin-cenzo criteria [2], need to be satisfied in order to beable to harness the power of quantum physics to ex-tract powerful information processing, and while mostof these criteria have been satisfied to a reasonable levelby some prospect for the implementation of a quantuminformation processing device [3], no such prospect hasyet been able to fulfill to a sufficient level all of these cri-teria at once. In particular, one criterion that not manyprospects seem to be able to fulfill is that of scalability,the ability of the model to remain coherent for quantumsystems with a large number of qubits.

Solid-state quantum registers seem to emerge as amedium that would allow for such scalability, while stillbeing able to satisfy the other criteria [3]. However, asfor any physical implementation of quantum computing,it suffers from noise, dissipation processes mainly in theform of decoherence and relaxation [4]. To be able touse the power of quantum computation, we must be ableto eliminate this noise and maintain quantum informa-tion throughout the processing. A general framework fordoing so is quantum error correction (QEC) [5, 6]. Itis known that by encoding a single logical qubit on suf-ficiently many physical qubits, we can correct arbitraryerrors on a single physical qubit [5]. But what happenswith more realistic multi-qubit errors?

In order to perform a quantum error correction, ad-ditional qubits are needed. However, these additionalqubits can lead to a much higher rate of decoherence(even superdecoherence) [7, 8]. The implementation ofthe three-qubit repetition code has already been simu-lated for a cavity-QED setup [9], showing a significantbut limited improvement for the preservation of quantum

information. Here, we consider a realistic noise model forN charge qubit quantum registers [8], and simulate thisnoise model on encoded qubits to evaluate the efficiencyof a QEC scheme. The quantum error correcting code(QECC) used is the 5 qubit DiVincenzo-Shor code [10].

We structure this work as follows: we first describe therepresentation for quantum systems we will be using, aswell as that for the noise model. We then further analyzethe physical model we study, and review the QECC weimplement within this model. Section VI then containsthe core of this work, that is, a description of the algo-rithm used for the simulation, followed by a discussion ofthe results obtained.

II. REPRESENTATION OF QUANTUMSYSTEMS

We will be considering the basic unit of quantum infor-mation: the qubit. A qubit can be implemented by anytwo level quantum system, and so can be represented by aunit length vector in a two-dimensional complex Hilbertspace, up to an equivalence relation for global phase. Ifwe consider some prefered orthonormal basis states |0〉and |1〉, a single qubit can be parametrized by 2 param-eters θ ∈ [0, π] and φ ∈ [0, 2π):

|ψ〉 = cos

(θ

2

)|0〉+eiφ sin

(θ

2

)|1〉 . (II.1)

This is the Bloch sphere representation, for pure qubitsystems.

When considering multiple interacting qubit systems,the overall system is described by the tensor product ofthe component Hilbert spaces. In general, once two sys-tems have interacted, a complete description of the wholesystem cannot be obtained by a description of each sub-system: this is entanglement. Much of the power of quan-tum information processing comes from entanglement, itis a uniquely quantum resource. Note that this is alsorelated to the difficulty of simulating a multi qubit quan-tum system: as a N qubit quantum system cannot be

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completely described by each of the N two dimensionalsubsystems, it must be described by a 2N dimensionalsystem, so that the memory required to store the infor-mation about a N qubit quantum system scales exponen-tially with N , and so do the time required to simulateoperations on this system.

This also raises the question of the description of sub-systems of entangled systems. It turns out that the den-sity operator formalism [11] can solve this issue. In thisformalism, a state ψ is represented by a density matrixρψ = |ψ〉〈ψ| instead of by a state vector |ψ〉. Then, ifwe are only interested in subsequent evolution of a par-ticular subsystem of the state ψ, we can do a partialtrace [11] over the other irrelevant subsytems and onlyevolve the reduced density matrix of the subsystem ofinterest. In this way, we obtain the right outcome statis-tic on this subsystem while not having to evolve a high-dimensionality state as required to describe the wholesystem completely. This density operator formalism alsocomes in handy for the description of statistical ensem-bles {pi, |ψi〉}, where the system is prepared in the state|ψi〉 with probability pi. The corresponding density op-erator is ρ =

∑i pi|ψi〉〈ψi|, and here also the evolution of

this state leads to the right measurement statistics.

A closely related subject is that of quantum noise. Asa quantum system cannot ever be completely isolatedfrom its environment, they interact and then become en-tangled. So, even though the global system-environmentpair undergoes unitary evolution, the main system doesnot, and this generates quantum noise. This is well de-scribed by the quantum operation formalism [11]. Twotypes of noise in which we will be mostly interested willbe decoherence and relaxation, and we will see that theseare closely related to generalized dephasing channels [12]and amplitude damping channels respectively [11, 13].

Since the quantum system of interest will be subjectto noise, we want a way to quantitatively describe howclose it will remain to the initial quantum informationwe wished to preserve. A good measure of this is thefidelity [11, 14], and for a pure input state |ψ〉 evolvingto a mixed state ρ, the fidelity is given by

F (|ψ〉, ρ) =√〈ψ | ρ |ψ〉. (II.2)

III. QUANTUM NOISE

The two main types of noise in which we will be in-terested are decoherence, which correspond to a loss ofquantum coherence without loss of energy, and relax-ation, which correspond to a loss of energy. We willsee that these types of noise are closely related to well-studied channels: generalized dephasing channels [12, 15]and amplitude damping channels [11, 13]

A. Generalized dephasing channels

Generalized dephasing channels correspond to physi-cal processes in which there is loss of quantum coherencewithout loss of energy. That is, there exist a preferredbasis, called the dephasing basis, such that pure statesin that basis are transmitted without error, but pure su-perposition get mixed, and quantum information is lostto the environment.

If we let A be the input system with orthonormal basis

{|i〉A}, B be the output system with orthonormal basis

{|i〉B}, and E be the environment with normalized, not

necessarily orthogonal states {|ξi〉E}, then an isometricmap from the input system to the output-environmentsystem is given by

UA→BE =∑i

|i〉B |ξi〉E 〈i |A . (III.1)

Then, starting with an input state ρA, we get the outputstate σB by first applying the isometry UA→BE , thentracing over E:

σB = TrE(UρAU†) =∑i,j

〈i | ρA |j〉 |i〉〈j|B〈ξj |ξi〉. (III.2)

As we can see, in this representation, the output corre-sponds to the Hadamard product between the (〈i | ρA |j〉)input matrix and the (〈ξi|ξj〉†) decoherence matrix. Also,since the |ξi〉 are normalized, 〈ξi|ξi〉 = 1, the diagonalterms are left unchanged:

〈i |σB |i〉 = 〈i | ρA |i〉 . (III.3)

These diagonal terms correspond to the probability ofmeasuring the quantum state in the corresponding de-phasing basis state, while the off-diagonal terms are co-herence (phase) terms. For these off-diagonal terms, theCauchy-Schwarz inequality (|〈ξi|ξj〉|2 ≤ 〈ξi|ξi〉〈ξj |ξj〉 =1) tells us that the output terms must be smaller thanor equal to the input terms: | 〈i |σB |j〉 | ≤ | 〈i | ρA |j〉 |.In the special case where the {|ξi〉E} are also orthogonal,we obtain a completely dephasing channel, which killsall off-diagonal terms, 〈i |σB |j〉 = 0 if i 6= j. For a sub-sequent measurement in the dephasing basis, this yieldsa classical statistical output with no quantum coherenceterms: σB =

∑i 〈i | ρA |i〉 |i〉〈i|B .

B. Amplitude damping channels

Amplitude damping channels correspond to physicalprocesses in which there is a loss of energy from thequantum system of interest to the environment. That is,there is a probability that a quantum state passes from ahigher energy excited state to a lower energy state. Forthe two-dimensional quantum systems in which we aremostly interested, this corresponds to a probability of

3

passing from the excited state to the ground state, whilethe ground state is left unaffected. In the quantum op-eration formalism, this channel has operation elements{E0, E1}:

E0 =

(1 00√

1− γ

), E1 =

(0√γ

0 0

), (III.4)

where the matrix representation is given in the energyeigenbasis, and the parameter γ corresponds to the prob-ability of decay. Then, for some input state ρA, we getan output state σB :

σB =∑i=0,1

EiρAE†i . (III.5)

If we represent the action of the channel as E , i.e.E(ρA) = σB , we can extend this definition to multi-qubitsystems. Making the assumption that energy lost foreach two-dimensional subsystem is independent, a ten-sor product of this channel, E⊗n, corresponds to a loss ofenergy in each subsystem independently.

IV. DECOHERENCE AND RELAXATION INCHARGE QUBIT REGISTERS

Solid state quantum systems represent an interest-ing prospect for the physical implementation of scalablequantum devices [3]. A lot of research goes into imple-mentation of superconducting qubits [16, 17]. Anotherimportant avenue of research are quantum dots, in whichtwo level quantum systems can be implemented in eitherthe spin or position degree of freedom of the electrontrapped in the dots. While spin quantum dots have alonger decoherence time [18, 19], spins are also more dif-ficult to manipulate. In charge quantum dots, where thebasis states correspond to the position of the electronin either of two adjacent quantum dots, the coupling isstrong and hence fast manipulation is possible, but thisalso leads to shorter decoherence times. We will considersuch a system, which offers an interesting playground forcurrent technology research [20, 21]. Moreover, the de-coherence for N qubit quantum registers, which can beseen in FIG. 1, has already been computed by Ischii et al.[8], and it is this model which is used in this simulation.

It is found that for these charge quantum dots, the de-coherence of the quantum system can be represented asa generalized dephasing channel in the canonical basis ofthe charge qubits, corresponding to the position eigen-states. Indeed, pure states in that basis are perfectlytransmitted, while pure superpositions become noisy. Itis then possible to compute the evolution of the quantumregister by taking the Hadamard product of the inputdensity matrix with a decoherence matrix computed in[8], thus getting the decohered output state. We choseas physical timescale unit ω0, which corresponds to theupper cut-off frequency. In charge qubits in GaAs wetypically have

ω0 = 0.2× 10−10s. (IV.1)

FIG. 1: Physical model for the 5-qubit quantum register. Thequbit geometry is inspired from recent experiments on coher-ent level mixing in vertical quantum dots [22].

This leads to the decay of the off-diagonal elements ofthe N-qubit density matrix. The structure of this decayis non-trivial and is shown in FIG. 2. In general, thisleads to an error, which cannot be reduced to a singlequbit error.

5 10 15 20 25 30

5

10

15

20

25

30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

5 10 15 20 25 30

5

10

15

20

25

30

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

T=10mK; t=0.1ωc T=1K; t=1ωc

)log( 1,1

00000|00000

)log( 32,1

00000|11111

)log( )log(

FIG. 2: (Color online) Matrix elements of the decay functiondensity matrix due to decoherence.

In addition, this decoherence model does not take intoaccount any relaxation the system might be subject to.We assume that the relaxation is independent of deco-herence, and also that energy loss for each qubit is inde-pendent. For each two dimensional quantum subsystem,if |0〉 and |1〉 correspond to the two position eigenstates,then the two energy eigenstates are |+〉 = 1√

2(|0〉+ |1〉)

for the ground state, and |−〉 = 1√2(|0〉− |1〉) for the

excited state. We can then compute the effect of relax-ation by operating a tensor product of amplitude damp-ing channels to each qubit in the register, where the op-eration elements (III.4) are in the {|+〉, |−〉} basis. Thedecay parameter γ is time (t) and temperature (T ) de-pendent,

γ = 1− exp(−t · T ), (IV.2)

setting kB = ~ = 1.

4

V. QUANTUM ERROR CORRECTION WITH APERFECT 5 QUBIT CODE

Since our quantum register is subject to noise, we wantto encode an input quantum state in such a way thatwe can preserve the quantum information even when thephysical system undergoes noise. A general frameworkfor doing this is quantum error correction (QEC) [5, 6],which encode a logical qubit containing the quantum in-formation on multiple physical qubits, and then usuallyby performing syndrome measurements on these physi-cal qubits, we can determine a certain set of errors, andapply the corresponding correction [11]. Many quantumerror correcting codes (QECC) have been found that cancorrect an arbitrary error on a single physical qubit, andit is known that the smallest such codes require 5 physicalqubits [23].

One of these 5 qubit codes is the DiVincenzo-ShorQECC [10]. The code words for the logical states{|0〉L, |1〉L} are [3]:

|0〉L =1

4

(|00000〉+ |11000〉+ |01100〉+ |00110〉

+ |00011〉+ |10001〉− |10100〉− |01010〉− |00101〉− |10010〉− |01001〉− |11110〉− |01111〉− |10111〉− |11011〉− |11101〉

)|1〉L =

1

4

(|11111〉+ |00111〉+ |10011〉+ |11001〉

+ |11100〉+ |01110〉− |01011〉− |10101〉− |11010〉− |01101〉− |10110〉− |00001〉− |10000〉− |01000〉− |00100〉− |00010〉

)with stabilizers [24]:

M0 = I ⊗ Z ⊗X ⊗X ⊗ ZM1 = Z ⊗ I ⊗ Z ⊗X ⊗XM2 = X ⊗ Z ⊗ I ⊗ Z ⊗X (V.1)

M3 = X ⊗X ⊗ Z ⊗ I ⊗ Z,

which leave the code invariant. The X and Z opera-tors are respectively the X and Z Pauli operators. Theencoding circuit can be seen in [3]. This is a perfectnon-degenerate code, meaning that all single qubit er-rors map to a different syndrome, as can be seen in TA-BLE I. To obtain a measurement of the error syndrome,we measure observables in the eigenbasis of each of thefour stabilizers in (V.1), and depending on the outcomeof these four measurements we apply the correspondingcorrection from TABLE I.

An alternative to these multi-qubit measurements [11]is to use ancillary qubits to conditionnally apply eachof the four stabilizers on a |+〉 state, thus recording thephase (M0-M3 have eigenvalues ±1). Measuring thesefour ancilla qubits in the {|+〉, |−〉} basis then gives thesame result as above without having to perform multi-qubit measurements. This syndrome measurement cir-cuit is given in [3].

M 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

EM I Z2 X0 Z3 X3 X1 Z4 Y3 Z1 X4 X2 Y2 Z0 Y1 Y0 Y4

TABLE I: Table of the different possible single qubit errorsEM , with their associated syndrome measurement output M ,which has binary expension M3M2M1M0.

This code is guaranteed to correct any single qubiterror, but it is not designed to correct multi-qubit errors,which our noise model will introduce. It will then beinteresting to see how well this code will perform for anapproximate correction of multi-qubit errors.

VI. QUANTUM ERROR CORRECTION IN ACHARGE QUBIT QUANTUM COMPUTER

The qubit model for physical implementation we con-sider in this simulation is a charge quantum dot. The twoprincipal sources of noise with this model are decoherenceand relaxation.

The effect of decoherence on a N -qubit register hadbeen computed in [8], where decoherence is caracteristicof a generalized dephasing channel [12], leaving compu-tational basis states unaffected.

The effect of relaxation correspond to a probabilisticloss of energy, where the excited state correspond to the|−〉 state, and the ground state correspond to the |+〉state. We make the assumption that this relaxation foreach qubit is independant, so that we implement thisnoise has a tensor product of an amplitude dampingchannel in the {|+〉, |−〉} basis, on each qubit.

To counter the effect of noise, we encode a logical qubitinto 5 qubits, using the DiVincenzo-Shor QECC, whichcan perfectly correct errors on a single qubit. However,the noise model used creates errors on multiple qubits,and so it is not obvious that QEC with the DiVincenzo-Shor code can extend the lifetime of our logical qubit.

Here, we verify the effect of such a quantum error cor-rection scheme, for different input states and differentfrequencies of correction. The algorithm used for thesimulation is presented first, and the results of the simu-lation are presented next.

A. Implementation of QEC algorithm

An overview of the algorithm can be seen in FIG. 3.First, the logical qubit is encoded into 5 physical qubitsusing the DiVincenzo-Shor QECC. This encoding is as-sumed to be perfect. We only consider pure input qubits,which are completely caracterized by two real parame-ters corresponding to their position on the Bloch sphere:θ ∈ [0, π], φ ∈ [0, 2π).

After encoding, the quantum register is subjected tonoise for a given period of time t, depending on the fre-quency of error correction. In our simulation, the de-

5

0

0

0

0

Encoding Noise

Quantum Error

Correction

10

0

0

0

0

LLenc10

Ancillary

Record Fidelity and Re-loop

FIG. 3: Overview of the algorithm for the simulation of quan-tum error correction of a solid-state quantum register.

coherence noise and the relaxation noise are applied suc-cessively, and we checked numerically that the order doesnot affect the result. The intensity of the noise dependson the temperature T of simulation, and the probabilityof relaxation is adjusted such that without error correc-tion, after a time ω0 the fidelity of the noisy state withrespect to the pure input state is the same as that ofdecoherence.

Quantum error correction is then performed. Here weassume perfect operation of each component of the QECalgorithm. However, instead of performing the measure-ment of the syndrome and then applying the correspond-ing correction (see TABLE I), we defer the measurementto the end of the circuit and perform conditional quan-tum correction [3, 11]. To do so, four ancillary qubits areused to record the syndrome measurements. It is pos-sible to avoid multi-qubit measurements by using theseancillary qubits, but the drawback is that we pass froma 5 qubit quantum system to a 9 qubit quantum system.This increases the difficulty of physical implementationbecause of the additionnal qubits, and also because everyround of QEC requires four fresh ancilla qubits. This alsoslows the simulation a lot, since we pass from a 25 = 32dimensional complex vector space to a 29 = 512 one, andthe matrix operations are accordingly scaled.

After performing the conditional QEC, these ancillaqubits are traced over, which corresponds to a measure-ment without a record of the outcome. This gives anoutput state which is a statistical ensemble correspond-ing to the different possible corrected states.

To perform the conditional quantum error correction,we implement a quantum circuit which would correct forthe different errors in TABLE I, corresponding to thevalue of the ancilla qubits. If we let |M〉 correspond tothe ancilla state with binary expansion M3M2M1M0, forM running from 0 to 15, and if EM is the corresponding

error, then the QEC circuit is

15∏M=0

(E†M ⊗ |M〉〈M |+ I5 ⊗ (I4 − |M〉〈M |)

), (VI.1)

where the order in the operator product is irrelevant sincethe terms commute.

Following the quantum error correction step, a recordof the fidelity of this output state compared to the inputstate is taken. We then reloop over the noise, quantumerror correction and fidelity recording steps, until the sumof all the noise time steps add up to the desired total timeof the simulation.

B. QEC performance

1. Decoherence

We can see on FIG. 4 the results of the simulationwhen our quantum register is subject only to decoher-ence. With a small frequency of error correction ofω−10 , the fidelity is already improved over the uncorrectedqubit. For a basis qubit, either |0〉 or |1〉, we do not havedecoherence, but the encoded state does suffer from de-coherence. The encoded basis states however do exhibitan interesting behavior when they are decohered, but noterror corrected. In fact, even though the noise takes the5 qubit state out of the code, when decoded we still getback the original basis state. For these basis states, wecould understand this behavior by noting that the codewords for the logical |0〉 and |1〉 contain different 5 qubitbasis states, as we can see in (V.1). But for pure super-position of these basis states, if we do not error correctthe encoded states, it also decoheres in a way such thatonce decoded, the same state as the unencoded decoheredstate is obtained, which is even more surprising.

2. Relaxation

We can see in FIG. 5 the results of the simulation whenwe only consider relaxation affecting our quantum regis-ter. This type of noise has a stronger temperature depen-dency and is stronger than decoherence for temperaturesabove 5 mK, at which point we obtain on average a sim-ilar drop in fidelity at 1 ω0. It is also harder to correctthan decoherence. We thus need a much higher frequencyof quantum error correction to keep the fidelity close to1 until we have reached a saturation in relaxation, thatis we have reached the lowest energy state for the un-encoded qubit. This low energy state is the only stateinvariant under action of the channel.

6

FIG. 4: Simulation of quantum error correction with onlydecoherence. The time between two successive QECs is∆t = ω0. For a total of 1000ω0, the top curve shows theresult of the QEC and the non-corrected 1qubit decoher-ence is shown in the bottom curve. The input qubit is|ψ〉 = cos

(12

)|0〉+ei sin

(12

)|1〉 and the temperature is as-

sumed to be 5 mK.

FIG. 5: Simulation of quantum error correction with onlyrelaxation: In the left graph the fidelity as a function of timeis shown for the same ∆t = ω0 as in FIG. 4. By reducing thetime interval between two QECs the fidelity remains close toone as seen in the right graph. For both graphs, the inputqubit is the same as in FIG. 4.

3. Simultaneous decoherence and relaxation

When combining the effect of decoherence and relax-ation, the frequency of error correction is on the order ofwhat is needed for relaxation, since relaxation is muchharder to correct than decoherence. Here too, when tak-ing into account decoherence and relaxation, we are stillable to maintain the fidelity arbitrarily close to one (seeFIG. 6). This tells us that even though it may require ahigh frequency of quantum error correction, the 5 qubitcode is able to correct a realistic multi-qubit error sys-tem.

C. Discussion

We have only considered noise in the N-qubit system.That is, an important assumption made for our simula-tion is that of a perfect operation of the QEC algorithm.

FIG. 6: 1 minus fidelity is shown as a function of time for avery short ∆t = 0.001ω0. The high-frequency QEC for jointdecoherence and relaxation enables the quantum informationto be preserved almost perfectly. The input qubit is the sameas in FIG. 4.

However, in a realistic fault-tolerant quantum compu-tation (FTQC) [25] , errors will also occur within theoperation of the gates and measurements, and this alsoduring the QEC steps. Similar concerns led some physi-cists to wonder if the error model adopted to get to thethreshold theorem [26, 27] for FTQC is physical enoughso that FTQC is possible at all [28]. Hence, such a simu-lation, with a realistic model for operations, would be animportant step toward determining if physical implemen-tations of quantum computers are possible. It is possiblethat these types of errors can be corrected in a similarmanner, but this is beyond the scope of this work.

The model considered introduces multi-qubit errors,which are much smaller than full single-qubit ones whenthe time evolution is very short. At the lowest order,we showed that these small multi-qubit errors can becorrected by single-qubit QEC codes. However, this in-creases the required QEC frequency, but it is certainlypossible to find other error correcting schemes optimizedfor multi-qubit errors, which would require a lower QECfrequency.

Also, since the decoherence time for charge qubits isshort, the frequency of QEC required to maintain goodfidelity for extended periods is too high to be feasibleexperimentally with current techniques. However, this isa proof of principle and it is reasonable to expect thatfor other implementations of qubits like spin qubits orsuperconducting qubits, where decoherence times can bemuch longer, similar performances would require a QECfrequency that is attainable experimentally.

VII. CONCLUSION

We have simulated the implementation of the 5 qubitDiVincenzo-Shor QECC for a charge qubit quantum reg-ister. The register was subjected to a realistic noise

7

model, consisting of decoherence and relaxation. Eventhough the QECC is designed to perfectly correct onlysingle qubit errors, an application of the QEC routinewith a high enough frequency was able to limit the effectof multi-qubit errors introduced by our noise model. Infact, by adjusting the frequency of error correction, wecan get the fidelity as close to one as possible, and forextended periods of time. When considering both typesof noise separately, relaxation showed to be harder to er-ror correct than decoherence, requiring higher frequency

of QEC for the same results.

Acknowledgement

M.H. acknowledges financial support from INTRIQ.D.T. acknowledges financial support from a NSERCUSRA for the duration of this work.

[1] D. Deutsch, Quantum theory, the Church-Turing princi-ple and the universal quantum computer, Proc. R. Soc.Lond. A, 400:97 (1985).

[2] D. P. DiVincenzo, The Physical Implementation ofQuantum Computation, Fortschr. Phys. 48, 771 (2000).arXiv:quant-ph/0002077v3

[3] M. Nakahara and T. Ohmi, Quantum Computing: FromLinear Algebra to Physical Realizations, (CRC Press,2008).

[4] U. Weiss, Quantum dissipative systems, (World Scientificpublishing co., third ed., 2008).

[5] P. W. Shor, Scheme for reducing decoherence in quantumcomputer memory, Phys. Rev. A 52, 2493 (1995).

[6] A. Steane, Simple Quantum Error Correcting Codes,Phys. Rev. A 54, 4741 (1996). arXiv:quant-ph/9605021v1

[7] G. M. Palma, K. A. Suominen and A. K. Ekert, Quantumcomputers and dissipation, Proc. Roy. Soc. Lond. A 452,567 (1996). arXiv:quant-ph/9702001v1

[8] B. Ischi, M. Hilke and M. Dube, Decoherence in a N-qubitsolid-state quantum register, Phys. Rev. B 71, 195325(2005). arXiv:quant-ph/0411086v2

[9] C. Ottaviani amd D. Vitali, Implementation of a three-qubit quantum error-correction code in a cavity-QEDsetup, Phys. Rev. A 82, 012319 (2010). arXiv:quant-ph/1005.3072v2

[10] D. P. DiVincenzo and P. W. Shor, Fault-Tolerant Er-ror Correction with Efficient Quantum Codes, Phys. Rev.Lett. 77, 3260 (1996). arXiv:quant-ph/9605031v2

[11] M. A. Nielsen and I. L. Chuang, Quantum Computa-tion and Quantum Information, (Cambridge Univ. Press,2000).

[12] I. Devetak and P. W. Shor, The capacity of a quantumchannel for simultaneous transmission of classical andquantum information, Commun. Math. Phys. 256, 287(2005) [ISI]. arXiv:quant-ph/0311131

[13] V. Giovannetti and R. Fazio, Information-capacity de-scription of spin-chain correlations, Phys. Rev. A 71,032314 (2005). arXiv:quant-ph/0405110v3

[14] B. W. Schumacher, Sending entanglement through noisyquantum channels, Phys. Rev. A 54, 2614 (1996).arXiv:quant-ph/9604023v1

[15] Kamil Bradler, Patrick Hayden, Dave Touchette, andMark M. Wilde, Trade-off capacities of the quantumHadamard channels, Phys. Rev. A 81, 062312 (2010).arXiv:quant-ph/1001.1732v2

[16] Y. Makhlin, G. Schon and A. Shnirman, Quantum-stateengineering with Josephson-junction devices, Rev. Mod.

Phys. 73, 357 (2001). arXiv:cond-mat/0011269v1[17] G. Wendin and V. S. Shumeiko, Superconducting Quan-

tum Circuits, Qubits and Computing, Prepared for Hand-book of Theoretical and Computational Nanotechnology(2005). arXiv:cond-mat/0508729v1

[18] J. M. Elzerman, R. Hanson, L. H. Willems van Beveren,B. Witkamp, L. M. K. Vandersypen and L. P. Kouwen-hoven, Single-shot read-out of an individual electron spinin a quantum dot, Nature 430, 431 (2004). arXiv:cond-mat/0411232v2

[19] J.R. Petta, A.C. Johnson, J.M. Taylor, E.A. Laird, A.Yacoby, M.D. Lukin, C.M. Marcus, M.P. Hanson andA.C. Gossar, Preparing, manipulating, and measuringquantum states on a chip, Physica E 35, 251 (2006).

[20] G. Shinkai, T. Hayashi, Y. Hirayama and T. Fuji-sawa, Controlled resonant tunneling in a coupled double-quantum-dot system, Appl. Phys. Lett. 90 103116 (2007).

[21] T. Hayashi, T. Fujisawa, H. D. Cheong, Y. H. Jeong, andY. Hirayama Coherent Manipulation of Electronic Statesin a Double Quantum Dot, Phys. Rev. Lett. 91, 226804(2003). arXiv:cond-mat/0308362v1

[22] C. Payette, S. Amaha, G. Yu, J. A. Gupta, D. G. Aust-ing, S. V. Nair, B. Partoens, and S. Tarucha, Coherentlevel mixing in dot energy spectra measured by magne-toresonant tunneling spectroscopy of vertical quantum dotmolecules, Phys. Rev. B 81, 245310 (2010).

[23] R. Laflamme, C. Miquel, J. P. Paz and W. H. Zurek,Perfect quantum error correction code, Phys. Rev. Lett.77, 198 (1996). arXiv:quant-ph/9602019v1

[24] D. Gottesman, A Class of Quantum Error-CorrectingCodes Saturating the Quantum Hamming Bound, Phys.Rev. A 54, 1862 (1996). arXiv:quant-ph/9604038v2

[25] P. W. Shor, Fault-tolerant quantum computation, 37thSymposium on Foundations of Computing, IEEE Com-puter Society Press, 1996, pp. 56-65. arXiv:quant-ph/9605011v2

[26] E. Knill, R. Laflamme, Concatenated Quantum Codes,arXiv:quant-ph/9608012v1.

[27] D. Aharonov, M. Ben-Or, Fault Tolerant Quantum Com-putation with Constant Error, Proceedings of the 29thAnnual ACM Symposium on Theory of Computing,1997, pp. 176-188. arXiv:quant-ph/9611025v2

[28] M. I. Dyakonov, Is Fault-Tolerant Quantum ComputationReally Possible?, Future Trends in Microelectronics. Upthe Nano Creek, Wiley (2007), pp. 4-18. arXiv:quant-ph/0610117v1