5-Qubit Quantum Error Correction in a Charge Qubit Quantum Computer

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

Dave Touchette, Haleemur Ali and Michael Hilke Department of Physics, McGill University, Montreal, Québec, H3A 2T8, Canada (Dated: October 18, 2010) We implement the DiVincenzo-Shor 5 qubit quantum error correcting code into a solid-state quantum 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 of the density matrix for this multi-qubit system and perform regular quantum error corrections. The performance of the error correction in this realistic system is found to yield an improvement of the fidelity. The fidelity can be maintained arbitrarily close to one by sufficiently increasing the frequency of error correction. This opens the door for arbitrarily long quantum computations.

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

I. INTRODUCTION information. Here, we consider a realistic noise model for N charge qubit quantum registers [8], and simulate this The task of building a quantum computer, allowing noise model on encoded qubits to evaluate the efficiency for massive quantum parallelism [1], long-lived quantum of a QEC scheme. The quantum error correcting code state superposition and entanglement, is one of the big (QECC) used is the 5 qubit DiVincenzo-Shor code [10]. challenge of twenty-first century physics. It requires pre- We structure this work as follows: we first describe the cise maintenance and manipulation of quantum systems representation for quantum systems we will be using, as at a level far from anything that has ever been done be- well as that for the noise model. We then further analyze fore. Many criteria, nicely summarized by the DiVin- the physical model we study, and review the QECC we cenzo criteria [2], need to be satisfied in order to be implement within this model. Section VI then contains able to harness the power of quantum physics to ex- the core of this work, that is, a description of the algo- tract powerful information processing, and while most rithm used for the simulation, followed by a discussion of of these criteria have been satisfied to a reasonable level the results obtained. by some prospect for the implementation of a quantum information processing device [3], no such prospect has II. REPRESENTATION OF QUANTUM yet been able to fulfill to a sufficient level all of these cri- SYSTEMS teria at once. In particular, one criterion that not many prospects seem to be able to fulfill is that of scalability, the ability of the model to remain coherent for quantum We will be considering the basic unit of quantum infor- systems with a large number of qubits. mation: the qubit. A qubit can be implemented by any two level quantum system, and so can be represented by a Solid-state quantum registers seem to emerge as a unit length vector in a two-dimensional complex Hilbert medium that would allow for such scalability, while still space, up to an equivalence relation for global phase. If being able to satisfy the other criteria [3]. However, as we consider some prefered orthonormal basis states |0i for any physical implementation of quantum computing, and |1i, a single qubit can be parametrized by 2 param- it suffers from noise, dissipation processes mainly in the eters θ ∈ [0, π] and φ ∈ [0, 2π): form of decoherence and relaxation [4]. To be able to use the power of quantum computation, we must be able θ iφ θ arXiv:1010.3242v1 [quant-ph] 15 Oct 2010 to eliminate this noise and maintain quantum informa- |ψi = cos |0i +e sin |1i . (II.1) 2 2 tion throughout the processing. A general framework for doing so is quantum error correction (QEC) [5, 6]. It This is the Bloch sphere representation, for pure qubit is known that by encoding a single logical qubit on suf- systems. ficiently many physical qubits, we can correct arbitrary When considering multiple interacting qubit systems, errors on a single physical qubit [5]. But what happens the overall system is described by the tensor product of with more realistic multi-qubit errors? the component Hilbert spaces. In general, once two sys- In order to perform a quantum error correction, ad- tems have interacted, a complete description of the whole ditional qubits are needed. However, these additional system cannot be obtained by a description of each sub- qubits can lead to a much higher rate of decoherence system: this is entanglement. Much of the power of quan- (even superdecoherence) [7, 8]. The implementation of tum information processing comes from entanglement, it the three-qubit repetition code has already been simu- is a uniquely quantum resource. Note that this is also lated for a cavity-QED setup [9], showing a significant related to the difficulty of simulating a multi qubit quan- but limited improvement for the preservation of quantum tum system: as a N qubit quantum system cannot be 2 completely described by each of the N two dimensional A. Generalized dephasing channels subsystems, it must be described by a 2N dimensional system, so that the memory required to store the infor- Generalized dephasing channels correspond to physi- mation about a N qubit quantum system scales exponen- cal processes in which there is loss of quantum coherence tially with N, and so do the time required to simulate without loss of energy. That is, there exist a preferred operations on this system. basis, called the dephasing basis, such that pure states This also raises the question of the description of sub- in that basis are transmitted without error, but pure su- systems of entangled systems. It turns out that the den- perposition get mixed, and quantum information is lost sity operator formalism [11] can solve this issue. In this to the environment. formalism, a state ψ is represented by a density matrix If we let A be the input system with orthonormal basis A ρψ = |ψihψ| instead of by a state vector |ψi. Then, if {|ii }, B be the output system with orthonormal basis we are only interested in subsequent evolution of a par- {|iiB}, and E be the environment with normalized, not ticular subsystem of the state ψ, we can do a partial E necessarily orthogonal states {|ξii }, then an isometric trace [11] over the other irrelevant subsytems and only map from the input system to the output-environment evolve the reduced density matrix of the subsystem of system is given by interest. In this way, we obtain the right outcome statis- tic on this subsystem while not having to evolve a high- A→BE X B E A U = |ii |ξii hi | . (III.1) dimensionality state as required to describe the whole i system completely. This density operator formalism also comes in handy for the description of statistical ensem- Then, starting with an input state ρA, we get the output bles {pi, |ψii}, where the system is prepared in the state state σB by first applying the isometry U A→BE, then |ψii with probability pi. The corresponding density op- tracing over E: P erator is ρ = i pi|ψiihψi|, and here also the evolution of this state leads to the right measurement statistics. B A † X A B σ = TrE(Uρ U ) = hi | ρ |ji |iihj| hξj|ξii. (III.2) A closely related subject is that of quantum noise. As i,j a quantum system cannot ever be completely isolated from its environment, they interact and then become en- As we can see, in this representation, the output corresponds to the Hadamard product between the (hi | ρA |ji) tangled. So, even though the global system-environment † pair undergoes unitary evolution, the main system does input matrix and the (hξi|ξji ) decoherence matrix. Also, not, and this generates quantum noise. This is well de- since the |ξii are normalized, hξi|ξii = 1, the diagonal scribed by the quantum operation formalism [11]. Two terms are left unchanged: types of noise in which we will be mostly interested will hi | σB |ii = hi | ρA |ii . (III.3) be decoherence and relaxation, and we will see that these are closely related to generalized dephasing channels [12] These diagonal terms correspond to the probability of and amplitude damping channels respectively [11, 13]. measuring the quantum state in the corresponding de- Since the quantum system of interest will be subject phasing basis state, while the off-diagonal terms are co- to noise, we want a way to quantitatively describe how herence (phase) terms. For these off-diagonal terms, the 2 close it will remain to the initial quantum information Cauchy-Schwarz inequality (|hξi|ξji| ≤ hξi|ξiihξj|ξji = we wished to preserve. A good measure of this is the 1) tells us that the output terms must be smaller than fidelity [11, 14], and for a pure input state |ψi evolving or equal to the input terms: | hi | σB |ji | ≤ | hi | ρA |ji |. E to a mixed state ρ, the fidelity is given by In the special case where the {|ξii } are also orthogonal, we obtain a completely dephasing channel, which kills p all off-diagonal terms, hi | σB |ji = 0 if i 6= j. For a sub- F (|ψi, ρ) = hψ | ρ |ψi. (II.2) sequent measurement in the dephasing basis, this yields a classical statistical output with no quantum coherence B P A B terms: σ = i hi | ρ |ii |iihi| .

III. QUANTUM NOISE B. Amplitude damping channels

The two main types of noise in which we will be in- Amplitude damping channels correspond to physical terested are decoherence, which correspond to a loss of processes in which there is a loss of energy from the quantum coherence without loss of energy, and relax- quantum system of interest to the environment. That is, ation, which correspond to a loss of energy. We will there is a probability that a quantum state passes from a see that these types of noise are closely related to well- higher energy excited state to a lower energy state. For studied channels: generalized dephasing channels [12, 15] the two-dimensional quantum systems in which we are and amplitude damping channels [11, 13] mostly interested, this corresponds to a probability of 3 passing from the excited state to the ground state, while the ground state is left unaﬀected. In the quantum operation formalism, this channel has operation elements {E0,E1}: √ 1 0 0 γ E = √ ,E = , (III.4) 0 0 1 − γ 1 0 0 where the matrix representation is given in the energy eigenbasis, and the parameter γ corresponds to the probability of decay. Then, for some input state ρA, we get an output state σB:

B X A † FIG. 1: Physical model for the 5-qubit quantum register. The σ = Eiρ Ei . (III.5) qubit geometry is inspired from recent experiments on coher- i=0,1 ent level mixing in vertical quantum dots [22]. If we represent the action of the channel as E, i.e. E(ρA) = σB, we can extend this deﬁnition to multi-qubit systems. Making the assumption that energy lost for This leads to the decay of the oﬀ-diagonal elements of each two-dimensional subsystem is independent, a ten- the N-qubit density matrix. The structure of this decay sor product of this channel, E⊗n, corresponds to a loss of is non-trivial and is shown in FIG. 2. In general, this energy in each subsystem independently. leads to an error, which cannot be reduced to a single qubit error.

IV. DECOHERENCE AND RELAXATION IN CHARGE QUBIT REGISTERS

T=10mK; t=0.1ω log() T=1K; t=1ω c c Solid state quantum systems represent an interest- 0.45 30 0.02 30 0.4 ing prospect for the physical implementation of scalable 0.018 quantum devices [3]. A lot of research goes into imple- 25 0.016 25 0.35 0.014 0.3 20 20 mentation of superconducting qubits [16, 17]. Another 0.012 0.25 0.01 important avenue of research are quantum dots, in which 15 15 0.2 0.008 0.15 two level quantum systems can be implemented in either 10 0.006 10 0.1 0.004 5 5 the spin or position degree of freedom of the electron 0.002 0.05

trapped in the dots. While spin quantum dots have a 5 10 15 20 25 30 5 10 15 20 25 30 longer decoherence time [18, 19], spins are also more dif-  log(1,1)  log(1,32 ) ficult to manipulate. In charge quantum dots, where the 00000 | 00000 11111 | 00000 basis states correspond to the position of the electron in either of two adjacent quantum dots, the coupling is strong and hence fast manipulation is possible, but this FIG. 2: (Color online) Matrix elements of the decay function also leads to shorter decoherence times. We will consider density matrix due to decoherence. such a system, which offers an interesting playground for current technology research [20, 21]. Moreover, the de- In addition, this decoherence model does not take into coherence for N qubit quantum registers, which can be account any relaxation the system might be subject to. seen in FIG. 1, has already been computed by Ischii et al. We assume that the relaxation is independent of deco- [8], and it is this model which is used in this simulation. herence, and also that energy loss for each qubit is inde- It is found that for these charge quantum dots, the de- pendent. For each two dimensional quantum subsystem, coherence of the quantum system can be represented as if |0i and |1i correspond to the two position eigenstates, a generalized dephasing channel in the canonical basis of then the two energy eigenstates are |+i = √1 (|0i + |1i) 2 the charge qubits, corresponding to the position eigen- for the ground state, and |−i = √1 (|0i − |1i) for the states. Indeed, pure states in that basis are perfectly 2 excited state. We can then compute the effect of relax- transmitted, while pure superpositions become noisy. It ation by operating a tensor product of amplitude damp- is then possible to compute the evolution of the quantum ing channels to each qubit in the register, where the op- register by taking the Hadamard product of the input eration elements (III.4) are in the {|+i, |−i} basis. The density matrix with a decoherence matrix computed in decay parameter γ is time (t) and temperature (T ) de- [8], thus getting the decohered output state. We chose pendent, as physical timescale unit ω0, which corresponds to the upper cut-off frequency. In charge qubits in GaAs we γ = 1 − exp(−t · T ), (IV.2) typically have −10 ω0 = 0.2 × 10 s. (IV.1) setting kB = ~ = 1. 4

V. QUANTUM ERROR CORRECTION WITH A M 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

PERFECT 5 QUBIT CODE EM I Z2 X0 Z3 X3 X1 Z4 Y3 Z1 X4 X2 Y2 Z0 Y1 Y0 Y4

Since our quantum register is subject to noise, we want TABLE I: Table of the different possible single qubit errors to encode an input quantum state in such a way that EM , with their associated syndrome measurement output M, we can preserve the quantum information even when the which has binary expension M3M2M1M0. physical system undergoes noise. A general framework for doing this is quantum error correction (QEC) [5, 6], which encode a logical qubit containing the quantum in- This code is guaranteed to correct any single qubit formation on multiple physical qubits, and then usually error, but it is not designed to correct multi-qubit errors, by performing syndrome measurements on these physi- which our noise model will introduce. It will then be cal qubits, we can determine a certain set of errors, and interesting to see how well this code will perform for an apply the corresponding correction [11]. Many quantum approximate correction of multi-qubit errors. error correcting codes (QECC) have been found that can correct an arbitrary error on a single physical qubit, and it is known that the smallest such codes require 5 physical VI. QUANTUM ERROR CORRECTION IN A qubits [23]. CHARGE QUBIT QUANTUM COMPUTER One of these 5 qubit codes is the DiVincenzo-Shor QECC [10]. The code words for the logical states The qubit model for physical implementation we consider in this simulation is a charge quantum dot. The two {|0iL, |1iL} are [3]: principal sources of noise with this model are decoherence 1 |0i = |00000i + |11000i + |01100i + |00110i and relaxation. L 4 The effect of decoherence on a N-qubit register had + |00011i + |10001i − |10100i − |01010i been computed in [8], where decoherence is caracteristic − |00101i − |10010i − |01001i − |11110i of a generalized dephasing channel [12], leaving computational basis states unaffected. − |01111i − |10111i − |11011i − |11101i The effect of relaxation correspond to a probabilistic 1 |1i = |11111i + |00111i + |10011i + |11001i loss of energy, where the excited state correspond to the L 4 |−i state, and the ground state correspond to the |+i + |11100i + |01110i − |01011i − |10101i state. We make the assumption that this relaxation for − |11010i − |01101i − |10110i − |00001i each qubit is independant, so that we implement this − |10000i − |01000i − |00100i − |00010i noise has a tensor product of an amplitude damping channel in the {|+i, |−i} basis, on each qubit. with stabilizers [24]: To counter the effect of noise, we encode a logical qubit into 5 qubits, using the DiVincenzo-Shor QECC, which M0 = I ⊗ Z ⊗ X ⊗ X ⊗ Z can perfectly correct errors on a single qubit. However, M1 = Z ⊗ I ⊗ Z ⊗ X ⊗ X the noise model used creates errors on multiple qubits, M = X ⊗ Z ⊗ I ⊗ Z ⊗ X (V.1) and so it is not obvious that QEC with the DiVincenzo- 2 Shor code can extend the lifetime of our logical qubit. M3 = X ⊗ X ⊗ Z ⊗ I ⊗ Z, Here, we verify the effect of such a quantum error cor- which leave the code invariant. The X and Z opera- rection scheme, for different input states and different tors are respectively the X and Z Pauli operators. The frequencies of correction. The algorithm used for the encoding circuit can be seen in [3]. This is a perfect simulation is presented first, and the results of the simu- non-degenerate code, meaning that all single qubit er- lation are presented next. 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 the A. Implementation of QEC algorithm four stabilizers in (V.1), and depending on the outcome of these four measurements we apply the corresponding An overview of the algorithm can be seen in FIG. 3. correction from TABLE I. First, the logical qubit is encoded into 5 physical qubits An alternative to these multi-qubit measurements [11] using the DiVincenzo-Shor QECC. This encoding is as- is to use ancillary qubits to conditionnally apply each sumed to be perfect. We only consider pure input qubits, of the four stabilizers on a |+i state, thus recording the which are completely caracterized by two real parame- phase (M0-M3 have eigenvalues ±1). Measuring these ters corresponding to their position on the Bloch sphere: four ancilla qubits in the {|+i, |−i} basis then gives the θ ∈ [0, π], φ ∈ [0, 2π). same result as above without having to perform multi- After encoding, the quantum register is subjected to qubit measurements. This syndrome measurement cir- noise for a given period of time t, depending on the fre- cuit is given in [3]. quency of error correction. In our simulation, the de- 5

Record Fidelity and Re-loop error, then the QEC circuit is    0   1    0   1 enc L L  15 Y † 0 EM ⊗ |MihM| + I5 ⊗ (I4 − |MihM|) , (VI.1) 0 Encoding Noise M=0 0 Quantum Error where the order in the operator product is irrelevant since 0 Correction the terms commute. Ancillary 0 Following the quantum error correction step, a record 0 of the ﬁdelity of this output state compared to the input 0 state is taken. We then reloop over the noise, quantum error correction and ﬁdelity recording steps, until the sum of all the noise time steps add up to the desired total time of the simulation.

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

1. Decoherence coherence noise and the relaxation noise are applied suc- cessively, and we checked numerically that the order does We can see on FIG. 4 the results of the simulation not affect the result. The intensity of the noise depends when our quantum register is subject only to decoher- on the temperature T of simulation, and the probability ence. With a small frequency of error correction of −1 of relaxation is adjusted such that without error correc- ω0 , the fidelity is already improved over the uncorrected tion, after a time ω0 the fidelity of the noisy state with qubit. For a basis qubit, either |0i or |1i, we do not have respect to the pure input state is the same as that of decoherence, but the encoded state does suffer from de- decoherence. coherence. The encoded basis states however do exhibit Quantum error correction is then performed. Here we an interesting behavior when they are decohered, but not assume perfect operation of each component of the QEC error corrected. In fact, even though the noise takes the algorithm. However, instead of performing the measure- 5 qubit state out of the code, when decoded we still get ment of the syndrome and then applying the correspond- back the original basis state. For these basis states, we ing correction (see TABLE I), we defer the measurement could understand this behavior by noting that the code to the end of the circuit and perform conditional quan- words for the logical |0i and |1i contain different 5 qubit tum correction [3, 11]. To do so, four ancillary qubits are basis states, as we can see in (V.1). But for pure super- used to record the syndrome measurements. It is pos- position of these basis states, if we do not error correct sible to avoid multi-qubit measurements by using these the encoded states, it also decoheres in a way such that ancillary qubits, but the drawback is that we pass from once decoded, the same state as the unencoded decohered a 5 qubit quantum system to a 9 qubit quantum system. state is obtained, which is even more surprising. This increases the difficulty of physical implementation because of the additionnal qubits, and also because every round of QEC requires four fresh ancilla qubits. This also slows the simulation a lot, since we pass from a 25 = 32 2. Relaxation dimensional complex vector space to a 29 = 512 one, and the matrix operations are accordingly scaled. We can see in FIG. 5 the results of the simulation when After performing the conditional QEC, these ancilla we only consider relaxation affecting our quantum regis- qubits are traced over, which corresponds to a measure- ter. This type of noise has a stronger temperature depen- ment without a record of the outcome. This gives an dency and is stronger than decoherence for temperatures output state which is a statistical ensemble correspond- above 5 mK, at which point we obtain on average a sim- ing to the different possible corrected states. ilar drop in fidelity at 1 ω0. It is also harder to correct To perform the conditional quantum error correction, than decoherence. We thus need a much higher frequency we implement a quantum circuit which would correct for of quantum error correction to keep the fidelity close to the different errors in TABLE I, corresponding to the 1 until we have reached a saturation in relaxation, that value of the ancilla qubits. If we let |Mi correspond to is we have reached the lowest energy state for the un- the ancilla state with binary expansion M3M2M1M0, for encoded qubit. This low energy state is the only state M running from 0 to 15, and if EM is the corresponding invariant under action of the channel. 6

FIG. 4: Simulation of quantum error correction with only FIG. 6: 1 minus fidelity is shown as a function of time for a decoherence. The time between two successive QECs is very short ∆t = 0.001ω0. The high-frequency QEC for joint ∆t = ω0. For a total of 1000ω0, the top curve shows the decoherence and relaxation enables the quantum information result of the QEC and the non-corrected 1qubit decoher- to be preserved almost perfectly. The input qubit is the same ence is shown in the bottom curve. The input qubit is as in FIG. 4. 1 i 1 |ψi = cos 2 |0i +e sin 2 |1i and the temperature is as- sumed to be 5 mK. However, in a realistic fault-tolerant quantum computation (FTQC) [25] , errors will also occur within the operation of the gates and measurements, and this also during the QEC steps. Similar concerns led some physi- cists to wonder if the error model adopted to get to the threshold theorem [26, 27] for FTQC is physical enough so that FTQC is possible at all [28]. Hence, such a simulation, with a realistic model for operations, would be an important step toward determining if physical implementations of quantum computers are possible. It is possible that these types of errors can be corrected in a similar FIG. 5: Simulation of quantum error correction with only manner, but this is beyond the scope of this work. relaxation: In the left graph the fidelity as a function of time The model considered introduces multi-qubit errors, is shown for the same ∆t = ω0 as in FIG. 4. By reducing the time interval between two QECs the fidelity remains close to which are much smaller than full single-qubit ones when one as seen in the right graph. For both graphs, the input the time evolution is very short. At the lowest order, qubit is the same as in FIG. 4. we showed that these small multi-qubit errors can be corrected by single-qubit QEC codes. However, this increases the required QEC frequency, but it is certainly 3. Simultaneous decoherence and relaxation possible to find other error correcting schemes optimized for multi-qubit errors, which would require a lower QEC When combining the effect of decoherence and relax- frequency. ation, the frequency of error correction is on the order of Also, since the decoherence time for charge qubits is what is needed for relaxation, since relaxation is much short, the frequency of QEC required to maintain good harder to correct than decoherence. Here too, when tak- fidelity for extended periods is too high to be feasible ing into account decoherence and relaxation, we are still experimentally with current techniques. However, this is able to maintain the fidelity arbitrarily close to one (see a proof of principle and it is reasonable to expect that FIG. 6). This tells us that even though it may require a for other implementations of qubits like spin qubits or high frequency of quantum error correction, the 5 qubit superconducting qubits, where decoherence times can be code is able to correct a realistic multi-qubit error sys- much longer, similar performances would require a QEC tem. frequency that is attainable experimentally.

C. Discussion VII. CONCLUSION

We have only considered noise in the N-qubit system. We have simulated the implementation of the 5 qubit That is, an important assumption made for our simula- DiVincenzo-Shor QECC for a charge qubit quantum reg- tion is that of a perfect operation of the QEC algorithm. ister. The register was subjected to a realistic noise 7 model, consisting of decoherence and relaxation. Even of QEC for the same results. though the QECC is designed to perfectly correct only single qubit errors, an application of the QEC routine with a high enough frequency was able to limit the effect of multi-qubit errors introduced by our noise model. In Acknowledgement fact, by adjusting the frequency of error correction, we can get the fidelity as close to one as possible, and for extended periods of time. When considering both types M.H. acknowledges financial support from INTRIQ. of noise separately, relaxation showed to be harder to er- D.T. acknowledges financial support from a NSERC ror correct than decoherence, requiring higher frequency USRA for the duration of this work.

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