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Coherence-Preserving Quantum Bits


Coherence-Preserving Quantum Bits
Dave Bacon1,2 , Kenneth R. Brown1 , and K. Birgitta Whaley1
Departments of Chemistry1 and Physics2 , University of California, Berkeley 94704 (February 1, 2008) Real quantum systems couple to their environment and lose their intrinsic quantum nature through the process known as decoherence. Here we present a method for minimizing decoherence by making it energetically unfavorable. We present a Hamiltonian made up solely of two-body interactions between four two-level systems (qubits) which has a two-fold degenerate ground state. This degenerate ground state has the property that any decoherence process acting on an individual physical qubit must supply energy from the bath to the system. Quantum information can be encoded into the degeneracy of the ground state and such coherence-preserving qubits will then be robust to local decoherence at low bath temperatures. We show how this quantum information can be universally manipulated and indicate how this approach may be applied to a quantum dot quantum computer. PACS Numbers: 03.67.Lx,03.65.Yz, 03.67.-a

arXiv:quant-ph/0012018v2 5 Aug 2002

One of the most severe experimental di?culties in quantum information processing is the fragile nature of quantum information. Every real quantum system is an open system which readily couples to its environment. This coupling causes the quantum information in the system to become entangled with its environment, which in turn results in the system information losing its intrinsic quantum nature. This process is known as decoherence. Circumvention of this decoherence problem has been shown to be theoretically possible with the development of the theory of fault-tolerant quantum error correction [1]. The set of requirements to reach the threshold for such fault-tolerant quantum computation is, however, extremely daunting. In this Letter we present a quantum informatic method for suppressing the detrimental e?ects of decoherence, while at the same time allowing for robust manipulation of the quantum information, in the hope that this method will aid in breeching the threshold for robust quantum computation [2]. In the absence of coupling between a system and its environment, the system and environment have separate temporal evolutions determined by their individual energy spectra. When a small interaction (relative to these energy scales) is switched on between the two, the resulting evolution is dominated by pathways that conserve the energy of the unperturbed system plus environment (rotating wave approximation, see [3]). Under the assumption of such a perturbative interaction, energetics play a key role in determining the rate of decoherence processes. Such energy conserving decoherence has three possible forms: energy is supplied from the system to the environment (cooling), energy is supplied from the environment to the system (heating), or no energy is exchanged at all (non-dissipative). Thus, even when the environment is a heat bath at zero temperature, cooling and especially non-dissipative interactions can be a major source of decoherence. The spirit of our approach to reducing decoherence is 1

to force all reasonable decoherence mechanisms to be interactions which heat the system, such that at low bath temperatures decoherence is energetically suppressed. This is done by encoding into logical qubits which are the ground state of a particular engineered Hamiltonian. While all dissipative and dephasing processes act on the physical qubits, the only source of decoherence on the encoded qubits derives from non-energy conserving decoherence pathways, which are by de?nition perturbatively weak. In particular, we will show the existence of a degenerate collective ground state of pairwise interacting two-level systems (qubits), which possesses the property that any local operation on an individual physical qubit must take the system out of this collective ground state. Quantum information can be encoded into the degeneracy of this ground state, to make an encoded qubit that is protected from any local decoherence which cannot overcome the established energy gap. Collective spin operations.—Let Hn = (C2 )?n be (i) a Hilbert space of n qubits, and let sα be the αth Pauli spin operator acting on the ith qubit tensored (i) with identity on all other qubits. The sα satisfy (j) (k) (j) (j) (k) 1 [sα , sβ ] = iδjk ?αβγ sγ and {sα , sβ } = 2 δjk δαβ I + 2(1?δjk )sα sβ . We de?ne the kth partial collective spin operators on the n qubits, Sα = i=1 sα . The total (n) collective spin operators acting on all n qubits, Sα , form a Lie algebra L which provides a representation of the (n) (n) (n) Lie algebra su(2): [Sα , Sβ ] = i?αβγ Sγ . Thus L can be decomposed in a direct product of irreducible repren/2 sentations (irreps) of su(2), L ? J=0,1/2 nJ L2J+1 , k=1 where L2J+1 is the 2J + 1 dimensional irrep of su(2) which appears with a multiplicity nJ . If we let (Jd )α be the operators of the d dimensional irrep of su(2), then there exists a basis for the total collective spin opera(n) n/2 tors such that Sα = J=0,1/2 InJ ? (J2J+1 )α . Cor(k) k (i) (j) (k)

responding to this decomposition of Sα , the Hilbert space H can be decomposed into states |λ, Jn , m classi?ed by quantum numbers labeling the irrep, Jn , the degeneracy index of the irrep, λ, and an additional internal degree of freedom, m. A complete set of commuting operators consistent with this decomposition and providing explicit values for these labels is given by (n) Bα = {(S(1) )2 , (S(2) )2 , . . . , (S(n?1) )2 , (S(n) )2 , Sα } [4]. Therefore a basis for the entire Hilbert space is given by |J1 , J2 , . . . , Jn?1 , Jn , mα , with (S(k) )2 |J1 , . . . , Jn , mα = (n) Jk (Jk + 1)|J1 , . . . , Jn , mα and Sα |J1 , . . . , Jn , mα = mα |J1 , . . . , Jn , mα . The degeneracy index λ of a particular irrep having total collective spin Jn is completely speci?ed by the set of partial collective spin eigenvalues Jk , k < n: λ ≡ (J1 , . . . , Jn?1 ). This degeneracy is simply due to the (nJ ) di?erent possible ways of constructing a spin-Jn out of n qubits. In Fig. 1 we present a graphical method for understanding this degeneracy of the irreps. The internal quantum number mα is the total spin projection along axis α. The |λ, Jn , m states have a particular clean property for decoherence mechanisms which couple collectively to the system. Quantum information encoded into the degeneracy |λ of these states is immune to collective decoherence. This information inhabits a decoherence-free (noiseless) subsystem [5–7,4]. Non-collective or local errors can still adversely a?ect decoherence-free subsystems [8]. In this paper we consider the action of independent errors acting on a code derived from decoherence free states and we show that these errors can be suppressed by suitable construction of the energy spectum. The decoherence-free property of the encoded states is retained in our approach. However, the method we present here deals with independent errors: as such, it can be used to reduce these errors irrespective of the existence of collective decoherence. (n) Collective Hamiltonian.—The Hamiltonian H0 = ? ? (n) 2 (S ) has eigenvalues 2 Jn (Jn + 1), with correspond2 ing eigenstates |λ, Jn , mα . Thus the (possibly degenerate) ground state of such a Hamiltonian is given by the lowest Jn states for a particular n. For n even, these states have Jn = 0, and for n odd they have Jn = 1/2. (n) Furthermore, H0 can be constructed from two-qubit interactions alone: H0
(n) (n)

(n)

(de?ned for n > 1). On determines which ?nal step is taken in the addition from qubit n?1 to qubit n (Fig. 1). If the ?nal step from Jn?1 to Jn was taken by adding 1/2, 1 then the eigenvalue of On will be On = Jn?1 + 2 , while if it was taken by subtracting 1/2, then On = ?(Jn?1 + 1 ). 2 It is convenient to replace (S(n) )2 by On in our set of commuting operators, which can clearly be done while still maintaining a complete set. We can then replace the quantum number Jn by On , to obtain the basis (n) (n) |λ, On , mα . It is easy to verify that {On , sα } = Sα . (n) If we examine the e?ect of sα on the basis |λ, On , mα (where we have de?ned mα in the orientation correspond(n) ing to Sα ), we ?nd that
′ ′ (On + On ) λ, On , mα |s(n) |λ′ , On , m′ α α
′ = mα δλ,λ′ δOn ,On δmα ,m′ α

(1)

=

? 2

n (i) i=j=1 s

· s(j) +

3n 4 I

.

Thus we see that H0 is nothing more than the Heisenberg coupling s(i) · s(j) acting with equal magnitude between every pair of qubits (I is an irrelevant energy shift). (n) E?ect of single qubit operators.—H0 has a highly degenerate spectrum, with energies determined by Jn . To determine the e?ect of single qubit operations on these states ?rst consider the e?ect of a single qubit operation (n) (n) on the nth qubit, sα . Since [sα , (S(k) )2 ] = 0 for k < n, (n) we see that sα can not change the degeneracy index λ of 1 a state |λ, Jn , mα . Let On = ? 4 I + (S(n) )2 ? (S(n?1) )2 2

Thus we see that the only non-zero matrix elements oc′ ′ cur when On = On or On = ?On . From this it follows that the ?nal step in the paths of Fig. (1) can either ?ip sign or else must remain the same. Using the relation between the On and Jn bases, this results in the selec(n) tion rules ?Jn = ±1, 0 for sα acting on states in the |λ, Jn , mα basis. Note further that if we had choosen a basis with mβ instead of mα in Eq. (1) (β = α), the same selection rules would hold, but now the mα components (n) could be mixed by sβ . In [4] it was shown that the ex1 change operation Eij = 2 I + 2s(i) s(j) which exchanges qubits i and j modi?es only the degeneracy index λ of (n) (j) the |λ, Jn , mα basis. Because sα = Ejn sα Ejn , this (i) implies that any single qubit operator sβ can therefore give rise to mixing of both the spin projections mα , and of the degeneracy indices λ. These selection rules must be modi?ed for the Jn = 0 states. On = ?1 and mα = 0 for all Jn = 0 states and any transitions between these states will therefore have (n) ′ zero matrix element, i.e., λ, Jn = 0, mα |sα |λ′ , Jn = ′ 0, mα = 0. Thus the transitions ?J = 0 are forbidden (n) for Jn = 0, and sα must take Jn = 0 states to Jn = 1 (n) ′ states. Furthermore, since λ, Jn = 0, 0|sα |λ′ , Jn = 0, 0 = 0, the degeneracy index λ for Jn = 0 states is not a?ected by any single qubit operation. To summarize, we have shown that any single qubit (i) operation sα enforces the selection rules ?Jn = ±1, 0 with the important exception of Jn = 0 which must have ?Jn = +1. The degenerate Jn = 0 states are therefore a quantum error detecting code for single qubit errors [9,4], with the special property that they are also the ground state of a realistically implementable Hamiltonian [10]. (n) The system Hamiltonian H0 has a ground state, for n even, with the remarkable property that all single qubit (i) errors sα become dissipative heating errors. Supercoherence.—Fig. (1) shows that for an even num(n) ber of qubits the Jn = 0 ground state of H0 is degen-

erate. For n = 4 physical qubits the ground state is twofold degenerate [6,9]. This degeneracy cannot be broken by any single qubit operator, and single qubit operations must take the J4 = 0 states to J4 = 1 states, as described (n) above. The system Hamiltonian H0 has a ground state, for n even, with the remarkable property that all single (i) qubit errors sα become dissipative heating errors. We will call this robust ground state a supercoherent qubit. If each qubit couples to its own individual environment, we expect that the major source of decoherence for these ground states will indeed be the processes which take the system from J4 = 0 to J4 = 1. What kind of robustness should we expect for the supercoherent qubit? If the individual baths have a temperature T , then we expect the decoherence rate on the encoded qubit to scale at low temperatures as ≈ e?β? , where β = (kT )?1 . At low temperatures there will thus be an exponential suppression of decoherence. Harmonic bath example.—As an example of the expected supercoherence we consider a quite general model of 4 qubits coupling to 4 independent harmonic baths. The unperturbed Hamiltonian of the system and bath (4) ? ? h is H0 ? I + I ? 4 i=1 ki ? ωki aki aki where aki is the creation operator for the ith bath mode with energy h ? ωki . The most general linear coupling between each sys(i) tem qubit and its individual bath is 4 α sα ? i=1 ki ? (gi,α aki + gi,α a? i ). According to the selection rules dek scribed above we can write sα = Am,n? i,α
(i) (m,n)∈S

Ai,α

(m,n)

+

h.c., where takes states Jn = m to Jn = n (and acts on λ and mα in some possibly nontrivial manner), and S is the set of allowed transitions S = {(0, 1), (1, 2), (1, 1), (2, 2)}. In the interaction picture, after making the rotating-wave approximation [3], we ?nd V(t) =
i,α,ki ,(m,n)∈S
? +gi,α ei( h f (m,n)?ωki )t Ai,α ?

? ? gi,α e?i( h f (m,n)?ωki )t Ai,α

?

(m,n) ? aki

(m,n)?

aki (2)

where f (m, n) = n(n + 1) ? m(m + 1). Coupling to thermal environments of the same temperature, under quite general circumstances (Markovian dynamics, well behaved spectral density of ?eld modes) we are led to a master equation (see for example [3]) ?ρ = ?t γi,α
i,α,(m,n)∈S (m,n) (m,n)? (m,n) (m,n)? (m,n)

Li,α

(m,n)

[ρ] + γi,α

(n,m)

Li,α

(n,m)

[ρ], (3)

with Li,α [ρ] = ([Ai,α ρ, Ai,α ]+[Ai,α , ρAi,α ]). The only operators which act on the supercoherent (0,1) qubit are Ai,α . The relative decoherence rates satisfy γi,α ∝ n(T ) where n(T ) = [exp(β?) ? 1] is the thermal average occupation number. Thus we see, as predicted that the supercoherent qubit decoheres at a rate which decreases exponentially as kT decreases below ?. 3
(0,1) ?1

(m,n)

Finally, we note that there are additional, two-qubit errors on the system which can break the degeneracy of the supercoherent ground state. Such terms will arise in higher order perturbation theory and will result in a 2 reduced energy gap of g . These terms will produce de? 2 coherence rates O(g /?2 ) smaller than the O(g 2 ) single qubit decoherence rates obtained without supercoherent encoding. In the perturbative regime, g ? ?, this factor therefore represents the small but ?nite limit to the protection o?ered by supercoherence. Universal quantum computation.—In order to be useful for quantum computation, the supercoherent qubits should allow for universal quantum computation. Extensive discussion of universal quantum computation on qubits encoded in decoherence-free subsystems has been given in [4,9] (see also [11,12]) where it was shown that computation on these encoded states can be achieved by turning on Heisenberg couplings between neighboring physical qubits. This means that we need to add extra Heisenberg couplings to the the supercoherent Hamilto(4) nian H0 . For a single supercoherent qubit these additional Heisenberg couplings can be used to perform any SU(2) rotation, i.e., an encoded one-qubit operation. In the present scheme one would like this additional coupling to avoid destroying the energy gap which suppresses decoherence. This can be achieved if the strength of the additional couplings, δ, is much less than the energy gap, i.e., δ ? ?. The trade-o? between the decoherence rate and the speed of the one qubit operations can be quanti?ed by calculating the gate ?delity F ∝ δeβ(??δ) . F quanti?es the number of operations which can be done within a typical decoherence time of the system. For small δ the gates are slower, while for larger δ the gap is smaller, resulting in a tradeo?. F is maximized for δ0 = kT . At this maximum F is still exponentially enhanced for lower temperatures. In particular, F |δ=δ0 ∝ β ?1 eβ? . Of more concern for the present scheme is how to perform computation between two encoded supercoherent qubits. It can be shown that using only Heisenberg couplings, a nontrivial two encoded qubit gate cannot (4) be done without breaking the degeneracy of the H0 Hamiltonian on the two sets of four qubits. This can be circumvented by considering a joint Hamiltonian of (8) the eight qubits, H0 . This Hamiltonian has a ground state which is 14-fold degenerate, including the tensor (4) product states of the degenerate ground state of the H0 Hamiltonian. The universality constructions previously presented in [4,9] can then easily be shown to never leave the ground state of this combined system. Having shown how to perform quantum computation on the encoded qubits, it is also apparent that the supercoherent qubit will su?er decoherence when there is a lack of control of the Heisenberg interactions used either (n) in constructing H0 or in performing a computation.

Unless the magnitude of ?uctuations in the Heisenberg interaction is larger in comparison to the bath temperature, the resistance of the supercoherent qubit to local decoherence is, however, una?ected by these errors. Supercoherence, then, represents a method for eliminating single-qubit decoherence process when superior two-qubit Hamiltonian control is possible [13]. Separation of control and decoherence:—A question which naturally arises is how the supercoherent qubits di?er from encoding into a degenerate or nearly degenerate ground state of a single physical system. For example, one could encode information into the nearly degenerate hyper?ne levels of an atomic ground state. There are essentially two di?erences between such a scheme and the supercoherent qubit. The ?rst di?erence lies in the fact that the degenerate ground state of a single quantum system can interact with its environment in such a way that the coherence of the state is lost without the bath supplying energy to the system, i.e. non-dissipatively. However, such a mechanism cannot a?ect a supercoherent qubit, (i) because all sα interactions have been shown above to supply energy from the bath to the system. A second di?erence lies in the extent of e?ciency in manipulation of the system. If a nearly degenerate ground state is used for quantum computation, there is a tradeo? between the speed of a single qubit gate and the decoherence rate. The limit on the speed of a supercoherent gate is, on the other hand, related to the temperature of the bath, with an error rate per quantum operation that scales in an exponentially favorable fashion. Supercoherent qubits therefore obtain a separation between controlled manipulation and uncontrolled decoherence, by making the control mechanisms two-body interactions which the singlequbit local decoherence cannot a?ect. Implementation in quantum dot grids.—The technological di?culties in building a supercoherent qubit are daunting but we believe within the reach of present experiments. In particular, these coherence-preserving qubit states appear perfect for solid state implementations of a quantum computer using quantum dots [14]. Related decoherence-free encodings on 3-qubit states were recently shown to permit universal computation with the Heisenberg interaction alone in [12]. The main new requirement for the supercoherent encoding, which allows the additional exponential suppression of decoherence not naturally achieved in decoherence-free states, (4) (8) (4) is the construction of H0 and H0 . H0 can be implemented by a two dimensional array with Heisenberg (8) couplings between all four qubits. H0 poses a more severe challenge, since the most natural geometry for implementing this Hamiltonian is eight qubits on a cube with couplings between all qubits. Such structures should be possible in quantum dots by combining lateral and vertical coupling scheme. Finally, estimates of the strength of the Heisenberg coupling in the quantum dot implemen4

tations are expected to be on the order of 0.1 meV [14]. Thus we expect that at temperatures below 0.1 meV ≈ 1 K, decoherence should be supressed for such coupled dots by encoding into the supercoherent states proposed here.

Acknowledgements: We thank Guido Burkard, David DiVincenzo, Julia Kempe, Daniel Lidar, and Taycee Lyn for useful conversations. This work was supported in part by the NSA and ARDA under ARO contract/grant number DAAG55-98-1-0371. The work of KRB is supported by the Fannie and John Hertz Foundation.

[1] An entry to the quantum error correcting literature is J. Preskill, Proc. Roy. Soc. London Ser. A 454, 469 (1998). [2] D. Aharonov and M. Ben-Or, In Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, p. 176 (1997); D. Gottesman, PhD thesis, Calif. Inst. of Tech., Pasadena, CA (1997); A. Y. Kitaev, In Quantum Communication, Computing, and Measurement, A. S. Holevo, O. Hirota, and C. M. Caves editors, p. 181, Plenum Press, New York, (1997); E. Knill, R. La?amme, and W. H. Zurek, Science 279, 342 (1998). [3] M. Scully and M. S. Zubairy, Quantum Optics, (Cambridge University Press, Cambridge 1997). [4] J. Kempe, D. Bacon, D. A. Lidar, and K. B. Whaley Phys. Rev. A, 63, 042307 (2001) [5] Decoherence-free subsystems are a generalization of decoherence-free subspaces (which were ?rst presented in Ref. [6].) [6] P. Zanardi and M. Rasetti, Mod. Phys. Lett. B 11,1085 (1997) [7] E. Knill, R. La?amme, and L. Viola, Phys. Rev. Lett. 84, 2525 (2000) [8] D.A. Lidar, I.L. Chuang, and K.B. Whaley, Phys. Rev. Lett. 81, 2594, (1998); D. Bacon, D.A. Lidar, and K.B. Whaley, Phys. Rev. A 60, 1944 (1999) [9] D. Bacon, J. Kempe, D. A. Lidar, and K. B. Whaley, Phys. Rev. Lett. 85, 1758 (2000). [10] A. Y. Kitaev, “Fault-tolerant quantum computation by anyons”, LANL preprint quant-ph/9707021; J. P. Barnes, W. S. Warren, Phys. Rev. Lett. 85, 856 (2000). [11] D. A. Lidar, D. Bacon, K. B. Whaley, Phys. Rev. Lett. 82, 4556 (1999). [12] D. P. DiVincenzo, D. Bacon, J. Kempe, G. Burkard, and K. B. Whaley, Nature 408, 339 (2000). [13] It is possible to construct supercoherent systems where the computation degrees of freedom are seperate from the supercoherent-inducing degrees of freedom: D. Bacon, Ph.D. thesis, Univ. Calif. Berkeley, 2001. [14] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). G. Burkard, D. Loss, and D. P. DiVincenzo, Phys. Rev. B 59, 2070 (1999).

J 2 3/2 1 1/2 0

J =2 04 =2 0 4 =-2 04 =1 04 =-1 J =1 J =0
? 2?

1

2

3

4 n J4

m

λ



FIG. 1. Diagram showing formation of the |λ, Jn , m states. The degeneracy index λ of a given J irreducible representation can be found by counting the number of paths which start with a spin-1/2 particle and which build up a total spin of J using standard addition of angular momenta. Thus, each path in this ?gure starting from n = 1, J1 = 1/2 is in one-to-one correspondence with a degeneracy index λ of a given J irrep. On is the eigenvalue of On for the ?nal step of this pathway. The allowed ?J transitions are shown as double-ended (4) arrows between the energy levels of H0 (see text for de?nition). Shown on the right are the λ and m degeneracies of the J4 levels. The energy di?erence δ corresponding to a computation on the supercoherent qubit will split the λ degeneracy.

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