Perfect mirror transport protocol with higher dimensional quantum chains

Perfect mirror transport protocol with higher dimensional quantum chains Gerardo A. Paz-Silva,1 Stojan Rebi´c,1 Jason Twamley,1 and Tim Duty2 1

Centre for Quantum Computer Technology, Macquarie University, Sydney, NSW 2109, Australia 2 Department of Physics, University of Queensland, St Lucia, QLD 4072, Australia

arXiv:0806.0194v2 [quant-ph] 23 Jan 2009

A globally controlled scheme for quantum transport is proposed. The scheme works on a 1D chain of nearest neighbor coupled systems of qudits (finite dimension), or qunats (continuous variable), taking any arbitrary initial quantum state of the chain and producing a final quantum state which is perfectly spatially mirrored about the mid-point of the chain. As a particular novel application, the method can be used to transport continuous variable quantum states. A physical realization is proposed where it is shown how the quantum states of the microwave fields held in a chain of driven superconducting coplanar waveguides can experience quantum mirror transport when coupled by switchable Cooper Pair Boxes. PACS numbers: 03.67Hk, 03.67Lx, 05.50.+q

A fundamental issue in quantum information theory and the development of a quantum computers is the transfer of a quantum state through a reliable, high fidelity, method. Such methods are known as quantum wires, or q-wires. Q-wire design should avoid the detailed control of its components as this may introduce additional errors which corrupt the transport and typically implies challenges in implementation. Ideally such quantum transport should be capable of transporting entire entangled registers of quantum information with near perfect fidelity and operate with the minimum of external control. Recently interest in q-wires has grown considerably due to the advent of higher dimensional quantum systems (qudits or CV, continuous variables) as components in quantum processors. So far, the only proposals for a mirror transport involved qubit states [1] and generally required significant resources to achieve useful transport fidelity [2, 3]. In this Letter a new protocol is described that for the first time formulates and solves the problem of quantum mirroring in chains consisting of nearest neighbor coupled identical qudit systems or infinite dimensional qunat [Continuous Variable (CV)] systems. It is based on a cellular automata-like scheme for perfect mirror transport [4], spatially inverting a quantum state distributed along a chain via the application of homogenous single qubit pulses and Ising-like nearest neighbor interactions. Qudits and qunats often lead to features which have certain advantages over qubits [5, 6]. Fault tolerant quantum computation has been shown for qudits [7], and schemes for trapped ion qudit quantum computation have been developed [8]. One-way quantum computing via qudit cluster states is possible, as is the formulation of qudit decoherence free subspaces and topologically protected qudit memories [9]. Moreover, continuous variable (CV) quantum computation has been widely studied in recent years [10], while CV quantum transport has also attracted recent interest [11]. Our CV results below provide an alternative method for perfect quantum transport in CV chains. The aim is to develop a quantum mirror transport protocol for qudit and CV systems. The qubit case was considered previously [4]. In this Letter a scheme suitable for qudits and CVs is developed. Physical realization is proposed involving perfect spatial mirroring of continuous variables in Circuit-

QED [12], spatially mirroring a quantum state of microwave photons trapped in a chain of coplanar waveguides (CPWs), coupled via Cooper pair boxes (CPBs). Qudit mirror transport. It will prove useful to recall some basic properties of qudit quantum gates [13, 14]. A qudit is a state of a d−level system described by a d−dimensional Hilbert space Hd with basis {|ki; k = 0, 1, · · · , d − 1}. Genˆ d ≡ Pd−2 |j⊕1ihj| eralized Pauli operators are defined via X j=0 Pd−1 and Zˆd ≡ |jiζ j hj|, where ζd is a d−th root of unity j=0

d

ζdd = 1, and ⊕ is addition modulo d. As for qubits, the ˆ ij ≡ X ˆ i Zˆ j , with the commutation relation qudit basis is Ξ d d d j jk j ˆ ˆ Zˆ k = ζ Zˆ k X ˆ ˆd ˆd X d k and Xd = Zd = I. In the qudit case d d d the Hadamard gate P is replaced by a Fourier gate Fˆ , given −1/2 ka ˆ by F |ai = d |ki, and satisfying Fˆ 4 = ˆI and kζ 2 Fˆ |ai = |−ai (−a is taken modulo d). Basic gates needed for our protocol are the qudit analogues of the CNOT (i.e. SUM), CPHASE and SWAP gates. Suppressing the dimension d and indicating the (control, target) qudits as subscripts [13, 15], ˆCP HASE = U

d−1 X

n |ni hn|(1) ⊗ Zˆ(2) = Sˆ(12) ,

(1a)

ˆn = D ˆ (12) , |ni hn|(1) ⊗ X (2)

(1b)

n=0

ˆSU M (12) = U =

d−1 X

n=0 −1 ˆ Fˆ(2) UCP HASE(12) Fˆ(2)

,

ˆSW AP = D ˆ (12) Fˆ 2 D ˆ ˆ2 ˆ ˆ2 U (1) (21) F(1) D(12) F(2) ,

(1c) (1d)

Aˆ(y) denoting the operator Aˆ acting on qudit y. Decomposiˆ are very simtions (1d) and (1c) (obtained using Zˆ Fˆ = Fˆ X) ilar in form to those found for the corresponding qubit gates: SW AP = CN OT12 CN OT21 CN OT12 and CN OT12 = ˆ ˆ (ˆI ⊗ H)CP HASE(ˆI ⊗ H). Fig. 1 shows the simplified qudit SWAP gate when one of the inputs is |0i. It builds an N −qudit perfect mirror circuit [Fig 2(A)] by replicating the simplified two qudit SWAP gate in Fig.1(D) N − 1 times and adding redundant CPHASE and

2

F-2

|Ψ!

F-2

|0!

F-2

|Ψ!

(A)

|0!

|Ψ!

F-1

F-1

F-1

F-1

F-1

F-2 +

|0!

|Ψ!

|0!

F-1

F-1

F-1

F-1

F-1

F-2 +

+

+

+

F-1

F-1

F-1

F-1

F-1

F-2 +

|0!

|0!

F-1

F-1

F-1

F-1

F-1

F-2

|Ψ!

(B)

F-2

|0!

|Ψ!

F-1 F-1

|Ψ!

|0!

F F-1

|0! |Ψ!

(C)

F-1

(D)

F F

F F

-1

-1

|0!

|Ψ!

+

F

-1

|Ψ!

Figure 1: Simplification of the qudit SWAP gate when one of the inputs is the |0i state. (A) Full SWAP from (1d), (B) shaded gates are redundant for input |0i2 , (C) inserting Fˆ Fˆ −1 on either side of the target on CSUM gates, (D) simplification using (1c).

Fˆ gates, where |+i = Fˆ +1 |0i. Through the repeated applicaQN tions of SF , with the global qudit gates F ≡ j=1 Fˆ(j) , and QN −1 S ≡ j=1 Sˆ(j,j+1) , one can transport the unknown quantum state from one end of the qudit chain to the other [21]. As it stands, it would seem from Fig. 2(A) that perfect mirror inversion requires the particular initial state |Ψi⊗|0i⊗|+i · · ·⊗|0i. We now show that this is not the case. It was shown for the qubit case that the transport does not depend on the input states [4]. The proof relies on N +1 ˆl = X ˆl showing that (H S)N +1 X and (a) (N +1−a) (H S) l N +1 ˆ l N +1 ˆ Z(a) = Z(N +1−a) (H S) which implies that (H S) any N −qubit density matrix is spatially inverted after the action of (H S)N +1 . For qudits, it suffices to show the analogous relations, i.e. F

±2

F

(F

±2

−1

(F

±2 −1 ˆl = X ˆl S)N +1 X (F S)N +1(2a) , (a) (N +1−a) F

−1

S)

N +1

±2 −1 l l Zˆ(a) = Zˆ(N (F S)N +1(.2b) +1−a) F

We now develop the basic algebraic relations used to prove (2). These relations will be very important in a generalization from qudits to continuous variables, as showing that they also hold for the CV case immediately verifies the validity of the mirror circuit in that case. To simplify the notation, the overbar denoting the global pulse is dropped unless stated otherwise by a subscript i.e. F → Fˆ , but Fˆ(a) → Fˆ(a) . Direct calculation shows ±2 ˆ l ˆ ±2 −l Fˆ(a) Z(a) F(a) = Zˆ(a) ,

(3)

ˆ l = Zˆ l Fˆ(a) . The operator (1a) can be written and Fˆ(a) X (a) (a) P j i as Sˆ(m,n) = ij ζ −ij Zˆ(m) ⊗ Zˆ(n) , to obtain ˆl = X ˆ l Zˆ l ˆ Sˆ(a,a+1) X (a) (a) (a+1) S(a,a+1) , ˆl Sˆ(a,a+1) X (a+1)

=

ˆl ˆl ˆ X (a+1) Z(a) S(a,a+1)

,

l l ˆ S(a,a+1) , Sˆ(a,a+1) Zˆ(a) = Zˆ(a)

ˆ −1

(F

ˆX ˆl S) (a)

=

ˆl ˆ −l ˆ l ˆ −1 S) ˆ X (a−1) Z(a) X(a+1) (F

ˆ Zˆ l = X ˆ l (Fˆ −1 S) ˆ . (Fˆ −1 S) (a) (a)

(A)

(4a) (4b) (4c) , (4d) (4e)

To derive Eqs. (2) a graphical method [4] is used [Fig.

+

(B) 1

2

a

N-a+1

N+1 N+2

1 a

N-a+1

N N+1 N+1

Figure 2: (A) Perfect mirror inversion circuit for qudits. Although particular input states are shown, the mirror action is independent of ˆ l ) under repeated actions of the input. (B) Evolution of the Zˆ l (and X −1 ˆ F S. The horizontal axis gives the number applications of Fˆ −1 S, ˆ l ( ) while the vertical axis is the qubit position. The Zˆ l () and X operators evolve according to (4), (black shape indicating a negative power of the correponding operator). After N + 1 repetitions the Zˆ l operator has moved to its mirror spatial position along the chain but has suffered l → −l. Negative exponent is corrected by the final Fˆ −2 operation.

ˆ N +1 ˆl = X ˆ −l ˆ −1 S) ˆ N +1 X 2(B)] to obtain (Fˆ −1 S) (a) (N +1−a) (F ˆ N +1 Zˆ l = Zˆ −l ˆ −1 S) ˆ N +1 . It follows and (Fˆ −1 S) (a) (N +1−a) (F [via (3)] that extra final Fˆ ±2 corrects the exponent l → −l. Fig. 2(B) shows the evolution of an initial Zˆ l operator after the consecutive action of Fˆ −1 S. It is extended to N + 2 so ˆ l and Zˆ l can be fully appreciated. It is the evolution of both X ˆ d was never important to note that the property Zˆ d = ˆI = X used, thus allowing us to use the same method for the CV case which lacks this cyclic property. Continuous variable mirror transport:- For continuous variables [10] the same argument is used. The CV analogues ˆ ˆ of the generalised Pauli operators are X(q) ≡ ζ −qpˆ, Z(p) ≡ pˆ x i/~ ˆ Z(p) ˆ ζ where ζ = e , are non-commutative, X(q) = ˆ X(q), ˆ ζ −qp Z(p) and have the following action on the compuˆ tational basis (position eigenstates x ˆ |qi = q |qi): X(q) |si = ˆ |si = η sp |si. By combining X(q) ˆ ˆ |s + qi, Z(p) and Z(p), ˆ ˆ) can be the displacement operator D(α) = exp(αˆ a† − α∗ a formed. Given that any valid infinite dimensional density operator can be decomposed Rinto a sum over coherent states via the P −function [16], ρˆ = d2 αP (α) |αi hα|, and that |αi = ˆ ˆ p) = X(q) ˆ Z(p); ˆ D(α) |0i, the operators Ξ(q, q, p ∈ R2 form ∞ a basis for H . This shows that CV-mirror transport protocol can be used to mirror any initial state of the CV-chain. Following the literature on CV-gates [10], basic gates are

3 defined using the same notation as in qudit case. The position representation will be used for all operators. The Fourier √ −1 R 2 dy ζ 2xy/σ |yi, sattransformation is Fˆ (σ) |xi = (σ π) √ isfying Fˆ 4 = ˆI and Fˆ 2 |xi = |−xi upon setting σ = 2. The basic gates needed are Z ± ˆ± ˆ (2) (±x) = CN d (12) (5a) U = dx |xi hx|(1) ⊗ X SU M (12) Z ± ± ˆ UCP HASE = dx |xi hx|(1) ⊗ Zˆ(2) (±x) = Sˆ(1,2) (5b) ± d (12) Fˆ −1 Fˆ(2) CN (2)

ˆ± where the identity U holds CP HASE = again. The mirror circuit can then be built√by using the + version of the defined gates and taking σ = 2. Note that using the + version of these gates allows to define the circuit independent of the basis [17]. With this in mind the reader shouldn’t be surprised that the relations ˆ Fˆ ±2 = Z(−s), ˆ ˆ ˆ ˆ ˆ Fˆ Fˆ ±2 Z(s) Fˆ Z(s) = X(−s) Fˆ , Fˆ X(s) = Z(s) −1 ˆ −1 ˆ −1 ˆ −1 ˆ ˆ ˆ ˆ ˆ F X(s) = Z(−s)F , F Z(s) = X(s)F (6) are also valid in this CV case. Since the algebraic relations (4d), (4e) are the same, the same proof used for the qudit case is valid in the CV case. Alternatively, we can use the traditional representations of SUM and Fourier transform in CV [17], i.e. without using the representation (5b), where ˆSU M (12) = CN d (12) = the SUM gate can be represented as U exp (−iˆ q1 ⊗ pˆ2 ), which has the following actions: d (12) : X ˆ 1 (x) ⊗ 12 → X ˆ 1 (x) ⊗ X ˆ 2 (x) , CN ˆ 2 (x) → X ˆ 2 (x) , Zˆ1 (p) → Zˆ1 (p), X −1 1i ⊗ Zˆj (p) → Zˆ (p) ⊗ Zˆj (p) , i

(7a) (7b) (7c)

ˆ −1

ˆ → Z, ˆ Fˆ : X Zˆ → X . (7d) QN ˆ QN −1 ˆ With these and F ≡ j=1 F(j) , and S = j=1 S(j,j+1) , we −1 ˆ (a) → arrive at relations analogous to (4d) and (4e) F S : X −1 ˆ (a−1) Zˆ X ˆ ˆ ˆ X (a) (a+1) and Z(a) → X(a) , so 2

F (F

−1

ˆ (a) → X(q) ˆ (N +1−a) S)N +1 : X(q) ˆ (a) → Z(p) ˆ (N +1−a) Z(p)

1.5

ha † ai F ull ha † ai E f f ha † ai Ham

1

1 0.8 0.6 0.4

0.5

hb † bi F ull hb † bi E f f hb † bi Ham

0.2 0 0

50

τ

100

150

0 0

50

τ

100

150

Figure 3: Quantum evolution of the two CPW modes in the adiabatic ˆ − , taking |ψ(τ = 0)i = |α = 1, β = 0, 1i, (the two limit for H modes in coherent states |αi ⊗ |βi and CPB state is |1i), and taking ω0 = 15GHz, ωa/b = 3GHz, ga/b = 200MHz, γ = 15MHz, κa/b = 1MHz. Plots show the reduced mode dynamics of the full master equation (9b) [Full], the effective master equation (10a) [Eff], and subject only to the effective Hamiltonian (10b) [Ham], with τ = ωa t/2π, the nominal oscillation period of the CPW modes.

The Hamiltonian and the resulting master equation for the two CV modes (operators a ˆ, ˆb) and CPB (σ± operators) operating at the charge degeneracy point are [12] ˆ ± /~ = ωa a H ˆ† a ˆ + ωbˆb†ˆb h i ω0 ˆz − ga (ˆ a+a ˆ† ) ± gb (ˆb + ˆb† ) (ˆ σ+ + σ ˆ− ), (9a) + σ 2 γ  i ˆ κa κb ρ˜˙ = [H ˜] + L(ˆ σ− ) + L(ˆ a) + L(ˆb) ρ˜ (9b) ±, ρ ~ 2 2 2 ˆ ± is the dynamics for the case of sum or difference where H bias conditions, ωa/b are the mode frequencies of the resonators, ga/b are the couplings between the CPB and each CPW, ω0 is the CPB splitting, γ, κa/b are the decay rates for ˆ ρ ≡ 2A˜ ˆρAˆ† − {Aˆ† A, ˆ ρ˜}. the CPB and two CPWs, and L(A)˜ Assuming that CPB dynamics dominate all other time scales in (9), i.e. ω0 > ωa/b , ga/b , and γ > κa/b , the fast dynamics of the CPB can be eliminated [18]. The equation for ρ = TrCP B {˜ ρ}, describing dynamics of the two coupled CV modes is (for ga = gb ) X ˆ ef f , ρ] + ρ˙ = −i/~[H κj L(j)ρ + ηL(ˆ s)ρ , (10a) j=ˆ a,ˆ b

(8)

as desired. Thus (8) proves that the circuit shown in Fig. 2(A), with the analogous CV gates performs perfect quantum mirroring on a chain of CV systems. Circuit-QED quantum mirroring:- The above CV quantum mirror transport can be realised in Circuit-QED [12]. The goal is to spatially mirror quantum state of N −CV modes held in chain of superconducting coplanar waveguides (CPWs), nearest-neighbour-coupled by switchable Cooper Pair Boxes (CPBs). To this end, F and S ought to be executed, where for S, transformation exp (−iˆ q1 ⊗ qˆ2 ) needs to be generated. We consider the arrangement where the CPB coupling adjacent 1/4-wave CPWs can be switched between being biased by the difference or the sum of the voltages of the two CV modes.

ˆ ef f = ~ωa a H ˆ† a ˆ + ~ωbˆb†ˆb + ~χˆ s2 ,

(10b)

ˆa ± X ˆ b and X ˆ a/b are the position operators where sˆ = X of the two CV modes χ = g 2 ω0 /[(γ/2)2 + ω02 ], and η = (γ/2)/[(γ/2)2 + ω02 ]. Fig. 3 shows that this approximation is valid for several hundred oscillator periods. ˆP HASE(12) , consider To generate the unitary operator U the effective Hamiltonian (10b), with the mode frequencies and effective coupling strengths time dependent ωa (t) = ωb (t). Such rapid tuning of the resonator frequencies is ˆ 0 (t)ef f = achievable [19]. The canonical transformation H † †ˆ ˆ ˆ ˆ ˆ T Hef f (t)T generated by T (a, b) ≡ exp(−π[ˆ a b − ˆb† a ˆ]/4) decouples (10b) into two separate harmonic systems. The reˆ 0 (t) = ω+ (t)(ˆ sulting Hamiltonian is H a0 † a ˆ0 + ˆb0 †ˆb0 ) + ef f ˆ 20 0 , where ω+ (t) = ωa (t) with either X ˆ 20 for sum 2χ(t)X a a /b

4

C1(τ)× 10−3

0

−20

−40 0

1

2

1

2

τ/2π

3

4

5

3

4

5

C2(τ)

0.2 0 −0.2 −0.4 0

τ/2π

ˆ ¯a0 ≡ exp(iX ˆ a20 /10) for parameters Figure 4: Pulse trains to effect U q q ˆ ˆ ˆ ¯a0 ) = ¯a0 )|/ Tr(U ¯a0 † U ˆ †U of Fig. 3 to an accuracy of F = |Tr(U 99.9% [20]. Total time to effect desired unitary operator Ua0 is thus 50 oscillator cycles.

ˆ 20 for difference voltage biasing. voltage biasing in (9b) or X b ˆCP HASE Under this canonical transformation the operator U 2 0 ˆ ˆ ˆ also separates UCP HASE = exp(iXa0 ) exp(−iXb20 ). Further, the Fourier gates applied homogeneously to both modes a ˆ and ˆb transform to homogeneously applied Fourier gates on the modes a ˆ0 and ˆb0 . This gate is achieved via the natural evolution of the uncoupled CPW resonators. We describe ˆ0 how to generate the U CP HASE by showing how to generate ˆ 20 ): (i) numerically find ˆa0 = exp(iX the first component U a time dependencies for ω+ (t) and χ(t), resulting in generaˆ 0 while simultaneously ˆb0 acquires a known rotation of U a tion exp(−iθˆb0 †ˆb0 ); (ii) Switch to difference voltage biasˆ 0 , while ing, evolving via a similar set of pulses, to effect U b 0 a ˆ acquires an identical known rotation. These extra rotations are then absorbed into the preceding and succeeding Fourier gates in the overall application of the Fig. 2(A). To derive the required time dependent pulse we focus on the a ˆ0 dynamics 0 ˆ in Hef f and move to a new time variable τ = ω ¯ t, where ω+ (t) = ω ¯ (1+C1 (t)/50), χ(t) = χ(1+C ¯ 2 (t)/50), and write ˆ ¯ a0 (τ ) = [1 + C1 (τ )/50]ˆ ˆ 20 . C1 (τ ) H a0 † a ˆ0 + [ + C2 (τ )/50]X a and C2 (τ ) are control functions and  = 2χ/¯ ¯ ω ∼ 1×10−3 for the parameters of Fig. 3. To obtain pulse sequences which attain 99.9% accuracy in simulating Ua0 , we use GRAPE-based numerical optimization algorithms [20] in 50 cycles of the oscillator and with a Fock state truncation of 70 [22]. The pulses, displayed in Fig. 4, require tuning the CPW resonators by 2.4MHz and the CPB by 150MHz. It is clear that if one can generate larger detunings of the oscillators and CPB, the time required to accurately simulate Ua0 can be even shorter, thus allowing several mirror iterates to be executed before the resonator decay degrades the dynamics. In summary a protocol for perfect quantum mirroring of a chain of coupled qudits or CV systems was proposed. Such

capability will allow the quantum transport of entire entangled qudit or CV quantum states via quantum mirroring. The protocol uses only global applications of higher dimensional versions of the Hadamard and CPHASE gates. We anticipate that CV universal quantum computation can be achieved if the protocol is augumented with the capability of performing a conditional phase gate between the end two CV qubits in the chain. This would require a large cross-Kerr nonlinear interaction between these CV systems. Finally, it was shown how to implement CV quantum mirroring in Circuit-QED. We thank European Commission FP6 IST FET QIPC project QAP Contract No. 015848, DEST ISL Grant CG090188 (SR & JT) and Mazda Foundation for Arts and Science (GAPS). [1] [2] [3] [4] [5] [6] [7] [8] [9]

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

[22]

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