Maximally entangling tripartite protocols for Josephson phase qubits
aa r X i v : . [ qu a n t - ph ] A p r Maximally entangling tripartite protocols for Josephson phase qubits
Andrei Galiautdinov ∗ Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602
John M. Martinis † Department of Physics, University of California, Santa Barbara, California 93106 (Dated: October 28, 2018)We introduce a suit of simple entangling protocols for generating tripartite GHZ and W statesin systems with anisotropic exchange interaction g ( XX + Y Y ) + ˜ gZZ . An interesting exampleis provided by macroscopic entanglement in Josephson phase qubits with capacitive (˜ g = 0) andinductive (0 < | ˜ g/g | < .
1) couplings.
PACS numbers: 03.67.Bg, 03.67.Lx, 85.25.-j
I. INTRODUCTION
Superconducting circuits with Josephson junctions have attracted considerable attention as promising candidatesfor scalable solid-state quantum computing architectures. The story began in the early 1980’s, when Tony Leggettmade a remarkable prediction that under certain experimental conditions the macroscopic variables describing suchcircuits could exhibit a characteristically quantum behavior [1]. Several years later such behavior was unambiguouslyobserved in a series of tunneling experiments by Devoret et al. [2], Martinis et al. [3], and Clarke et al. [4]. It waseventually realized that due to their intrinsic anharmonicity, the ease of manipulation, and relatively long coherencetimes [5], the metastable macroscopic quantum states of the junctions could be used as the states of the qubits. Thatidea had recently been supported by successful experimental demonstrations of Rabi oscillations [6], high-fidelity statepreparation and measurement [7–12], and various logic gate operations [8–11, 13]. Further progress in developing aworkable quantum computer will depend on the architecture’s ability to generate various multiqubit entangled statesthat form the basis for many important information processing algorithms [14].In this paper we develop several single-step entangling protocols suitable for generating maximally entangled quan-tum states in tripartite systems with pair-wise coupling g ( XX + Y Y ) + ˜ gZZ . We base our approach on the ideathat implementing symmetric states may conveniently be done by symmetrical control of all the qubits in the system.This bears a resemblance to approaches routinely used in digital electronics: while an arbitrary gate (for example, a3-bit gate) can be made from a collection of NAND gates, it is often convenient to use more complicated designs withthree input logic gates to make the needed gate faster and/or smaller.The protocols developed in this paper may be directly applied to virtually any of the currently known superconduct-ing qubit architectures, two of which will be mentioned here. The first architecture is based on capacitively coupledcurrent-biased (CBJJ) Josephson junctions [12, 13, 15] whose dynamics is governed by the circuit Hamiltonian H = (cid:0) p + p + κp p (cid:1) / m + ( ~ / e ) X i =1 [ − I cos φ i − I i φ i ] , (1)with p i = m ˙ φ i , m = ( ~ / e ) ( C + C int ), κ = 2 C int / ( C + C int ). The other architecture involves inductively coupledflux-biased (FBJJ) junctions [16]. It is described by the Hamiltonian (for small Υ, see Ref. [17] for details) H = (cid:0) p + p (cid:1) / m + ( ~ / e ) X i =1 h − I cos φ i + ( eE / ~ ) ( φ i − π Φ i / Φ sc ) i + Υ E ( φ − π Φ i / Φ sc ) ( φ − π Φ i / Φ sc ) , (2)with p i = m ˙ φ i , m = ( ~ / e ) C , ω = 1 / √ LC , E = ~ ω / E C , E C = (2 e ) / C , Φ sc = h/ e , Υ = M/L . Whenreduced to computational subspace, in the rotating wave approximation, these Hamiltonians become H RWA = (1 / h ~ Ω · ~σ + ~ Ω · ~σ + g (cid:0) σ x σ x + σ y σ y (cid:1) + ˜ gσ z σ z i , (3) ∗ Electronic address: [email protected] † Electronic address: [email protected] with ˜ g = 0 (momentum-momentum coupling) and 0 < ˜ g/g < . ω/ π ≃
10 GHz and the Rabi frequencies Ω needed to implement various logic gates [17, 19] are on the order of thecoupling constant g/ π .
100 MHz. Thus, Ω /ω ∼ − , as required. II. THE GHZ PROTOCOLA. Triangular coupling scheme
In the rotating frame in the absence of coupling, the computational basis states | i , | i , | i , | i , | i , | i , | i , | i have the same effective energy E eff = 0. The pair-wise coupling, H int = (1 / X i,j =1 g (cid:0) σ ix σ jx + σ iy σ jy (cid:1) + ˜ gσ iz σ jz = g/ − ˜ g/ g gg − ˜ g/ gg g − ˜ g/ − ˜ g/ g gg − ˜ g/ gg g − ˜ g/ g/ , (4)(empty matrix elements are zero), partially lifts the degeneracy, which results in the energy spectrum E int = { g/ , g/ , g − ˜ g/ , g − ˜ g/ , − ( g + ˜ g/ , − ( g + ˜ g/ , − ( g + ˜ g/ , − ( g + ˜ g/ } , (5)with the corresponding H -eigenbasis H GHZ M H W M H rest ≡ {| i ⊕ | i} M {| W i ⊕ | W ′ i} M (cid:8) | Ψ i ⊕ | Ψ ′ i ⊕ | Ψ i ⊕ | Ψ ′ i (cid:9) , (6)where | W i = ( | i + | i + | i ) / √ , | W ′ i = ( | i + | i + | i ) / √ , | Ψ i = ( | i − | i ) / √ , | Ψ ′ i = ( | i − | i ) / √ , | Ψ i = ( | i + | i − | i ) / √ , | Ψ ′ i = ( | i + | i − | i ) / √ . (7)Since the coupling does not cause transitions within each of the degenerate subspaces (nor does it cause transitionsbetween different such subspaces), it is impossible to generate the | GHZ i = ( | i + | i ) / √ | i by direct application of H int . Instead, we must first bring the | i state out of the H GHZ subspaceby, for example, subjecting it to a local rotation R in such a way as to produce a state | ψ i that has both | i and | i components. That is only possible if all one-qubit amplitudes α , . . . , β in the resulting product state | ψ i = R | i = ( α | i + β | i ) ( α | i + β | i ) ( α | i + β | i ) are chosen to be nonzero, which means that in thecomputational basis the state | ψ i will have eight nonzero components.We now notice that in the H -basis, the three-qubit rotations X θ = X (3) θ X (2) θ X (1) θ = c is − i √ sc −√ cs is c −√ cs − i √ sc − i √ sc −√ cs c (1 − s ) is (1 − c ) −√ cs − i √ sc is (1 − c ) c (1 − s ) ⊕ (cid:18) c isis c (cid:19) ⊕ (cid:18) c isis c (cid:19) ,Y θ = Y (3) θ Y (2) θ Y (1) θ = c − s −√ sc √ cs s c √ cs √ sc √ sc √ cs c (1 − s ) s (1 − c ) √ cs −√ sc − s (1 − c ) c (1 − s ) ⊕ (cid:18) c s − s c (cid:19) ⊕ (cid:18) c s − s c (cid:19) , (8)where Y ( k ) θ = exp (cid:0) − iθσ ky / (cid:1) , X ( k ) θ = exp (cid:0) − iθσ kx / (cid:1) , k = 1 , ,
3, are block-diagonal, with c ≡ cos( θ/
2) and s ≡ sin( θ/ θ = π/
2, the corresponding 4 × H GHZ L H W subspace are X (4 × π/ = 1 √ i − i √ −√ i −√ − i √ − i √ −√ − − i −√ − i √ − i − , Y (4 × π/ = 1 √ − −√ √
31 1 √ √ √ √ − − √ −√ − . (9)This shows that Y π/ provides a convenient choice for R . We may thus start by generating the so-called symmetricstate, | ψ i sym = Y π/ | i = (1 / (cid:16) | GHZ i + p / | W i + | W ′ i ) (cid:17) ∈ H GHZ M H W . (10)The entanglement is then performed by acting on | ψ i sym with U int = exp ( − iH int t ), thus inducing a phase differencebetween the GHZ and W+W ′ components (this step works only for g = ˜ g , see Section IV), U int Y π/ | i = (cid:0) e − iα / (cid:1) (cid:16) | GHZ i + e − iδ p / | W i + | W ′ i ) (cid:17) , α = (3˜ g/ t, δ = 2 ( g − ˜ g ) t. (11)To transform to the desired GHZ state, we first diagonalize the X (4 × π/ and Y (4 × π/ operators to get the unimodularspectra λ X = n − e i ( π/ , − e − i ( π/ , e − i ( π/ , e i ( π/ o , λ Y = n − e − i ( π/ , − e i ( π/ , e i ( π/ , e − i ( π/ o , (12)corresponding to the X - and Y -eigenbases, X = (cid:0) | X i | X i | X i | X i (cid:1) ≡ Y (4 × π/ , Y = (cid:0) | Y i | Y i | Y i | Y i (cid:1) ≡ X (4 × π/ , which are formed by the columns of Y (4 × π/ and X (4 × π/ . Using the X -basis, we notice that both states | GHZ i = | X i + √ | X i , U int Y π/ | i = e − iα (cid:18) e − iδ | X i + 1 − e − iδ √ | X i (cid:19) , (13)belong to the same two-dimensional (nondegenerate) X -subspace spanned by | X i ⊕ | X i . Therefore, by performingan additional X π/ rotation we can transform U int Y π/ | i to X π/ U int Y π/ | i = e − iα e i ( π/ | GHZ i , (14)provided the entangling time is set to give | δ | = π , or, t GHZ = π/ | g − ˜ g | . Any other GHZ state ( | i + e iφ | i ) / √ Z -rotation applied to one of the qubits, as usual.The protocol may be compared to controlled-NOT logic gate implementations [17, 19] that used various sequencesCNOT = e − i ( π/ R U CNOT R , with (entangling) times t CNOT = T π/ g , 1 ≤ T < .
6. Thus, for ˜ g = 0, the entanglingoperation proposed here will be of same duration as the fastest possible CNOT.We conclude this section by noting that in its present form the GHZ protocol cannot be used to generate the Wstate. This can be seen by writing | W i = (cid:0) √ | X i + | X i ) − ( | X i + | X i ) (cid:1) / √
8, which shows that our XU int Y sequence does not result in a W since the final X π/ rotation cannot eliminate the | X i and | X i components. Also, | W i = (cid:16) √ i | Y i − | Y i ) − ( | Y i − i | Y i ) (cid:17) / √ , (15)and Y π/ U int Y π/ | i = e − iα − e − iδ i | Y i − | Y i ) − √ (cid:0) e − iδ (cid:1) | Y i − i | Y i ) ! / √ , (16)and thus no choice of δ will work for the Y U int Y sequence either. B. Linear coupling scheme
In the case of linear coupling, say, 1 ↔ ↔
3, the interaction Hamiltonian is given by H int = ˜ g g g − ˜ g g g g g − ˜ g g g g , E int = n ˜ g, ˜ g, ǫ (+) , ǫ (+) , ǫ ( − ) , ǫ ( − ) , , o , ε ( ± ) = ± q g + (˜ g/ − ˜ g/ , (17)with eigenbasis | i , | i , | W i (+) = C (+) (cid:16) | i + ( ǫ (+) /g ) | i + | i (cid:17) , | W ′ i (+) = C (+) (cid:16) | i + ( ǫ (+) /g ) | i + | i (cid:17) , | W i ( − ) = C ( − ) (cid:16) | i + ( ǫ ( − ) /g ) | i + | i (cid:17) , | W ′ i ( − ) = C ( − ) (cid:16) | i + ( ǫ ( − ) /g ) | i + | i (cid:17) , | Ψ i = ( | i − | i ) / √ , | Ψ ′ i = ( | i − | i ) / √ , (18)where C ( ± ) are normalizing constants. We have, | W i = A (+) | W i (+) + A ( − ) | W i ( − ) , A (+) = − ǫ ( − ) + gǫ (+) − ǫ ( − ) (cid:18) C (+) (cid:19) , A ( − ) = ǫ (+) − gǫ (+) − ǫ ( − ) (cid:18) C ( − ) (cid:19) , (19)and similarly for | W ′ i . Our GHZ sequence then leads to the entangled state U int Y π/ | i = (cid:0) e − iα / (cid:1) (cid:16) | GHZ i + p / (cid:16) e − iδ (+) A (+) h | W i (+) + | W ′ i (+) i + e − iδ ( − ) A ( − ) h | W i ( − ) + | W ′ i ( − ) i(cid:17)(cid:17) , (20)with α = ˜ gt , δ ( ± ) = ( ǫ ( ± ) − ˜ g ) t . Since t >
0, in order for the X π/ post-rotation to give a GHZ, we must restrict couplingto ˜ g = 0 and set the entangling time to t GHZ = π/ √ | g | . An alternative GHZ implementation for superconductingqubit systems with capacitive coupling has recently been considered in Refs. [21, 22]. There, individual qubits wereconditionally operated upon one at a time. III. THE W PROTOCOL
We now turn to the W protocol. Eq. (16) suggests that control sequence
Y U int Y may still give a W, provided aproper adjustment of i | Y i − | Y i and | Y i − i | Y i amplitudes is made by a physically acceptable change of system’sHamiltonian. In the context of Josephson phase qubits such modification can be achieved by adding local Rabi term(s)to H int , for instance, H Ωint = (Ω / (cid:0) σ x + σ x + σ x (cid:1) + H int = 12 g Ω Ω ΩΩ − ˜ g g g Ω ΩΩ 2 g − ˜ g g Ω ΩΩ 2 q g − ˜ g Ω ΩΩ Ω − ˜ g g g ΩΩ Ω 2 g − ˜ g g ΩΩ Ω 2 g g − ˜ g ΩΩ Ω Ω 3˜ g . (21)The energy spectrum then becomes E Ωint = (cid:8) ǫ (+) ± χ (+) , ǫ ( − ) ± χ ( − ) , − ǫ (+) , − ǫ (+) , − ǫ ( − ) , − ǫ ( − ) (cid:9) , with ǫ ( ± ) = g +˜ g/ ± Ω / χ ( ± ) = p ( g − ˜ g ) ± ( g − ˜ g )Ω + Ω . The (first two) eigenvectors are | Φ (+)1 , i = C (+)1 , (cid:16)h − − (2 / Ω) (cid:16) g − ˜ g ∓ χ (+) (cid:17)i | GHZ i + p / | W i + | W ′ i ) (cid:17) , (22)with normalizing constants C (+) k , k = 1 ,
2. After some algebra we find: U Ωint Y π/ | i = e − iα / (4 √ χ (+) ) h ( A/ Ω) (cid:16) ie i ( π/ | Y i − e − i ( π/ | Y i (cid:17) + ( √ B/ Ω) (cid:16) e − i ( π/ | Y i − ie i ( π/ | Y i (cid:17)i , (23)where A = (cid:16) g − ˜ g + Ω + χ (+) (cid:17) (cid:16) g − ˜ g + 2Ω − χ (+) (cid:17) − e − iδ (cid:16) g − ˜ g + Ω − χ (+) (cid:17) (cid:16) g − ˜ g + 2Ω + χ (+) (cid:17) ,B = (cid:16) g − ˜ g + Ω + χ (+) (cid:17) (cid:16) g − ˜ g − χ (+) (cid:17) − e − iδ (cid:16) g − ˜ g + Ω − χ (+) (cid:17) (cid:16) g − ˜ g + χ (+) (cid:17) , (24)and α = (cid:0) ǫ (+) + χ (+) (cid:1) t , δ = − χ (+) t . It is straightforward to verify that additional Y π/ rotation applied to thisstate produces a W (see Eqs. (12), (15)), Y π/ U Ωint Y π/ | i = [ − sgn ( g − ˜ g )] e − iα | W i , (25)provided we set t W = π/ √ | g − ˜ g | , Ω = − ( g − ˜ g ) / IV. ADDENDUM: ISOTROPIC HEISENBERG EXCHANGE g ( XX + Y Y + ZZ ) Maximally entangling protocols introduced in previous sections are singular in the limit ˜ g → g , which correspondsto the isotropic Heisenberg exchange interaction. Even though this limit is not met in superconducting qubits, forcompleteness, we briefly discuss it here.It is obvious that when g = ˜ g , the symmetric state Y π/ | i is an eigenstate of the interaction Hamiltonian.Consequently, the Heisenberg exchange does not cause transitions out of it, making the gate time divergent. Toperform single-step entanglement we break the symmetry of local rotations. For example, the GHZ state can begenerated by e − iα | GHZ i = e − i ( π/ σ z e − i ( π/ ( σ y − σ y ) U int e − i ( π/ ( σ y + σ y − σ y ) e − i ( π/ σ z | i , with α = − π/ t GHZ =(2 / × ( π/ g ). To generate the W state, we generalize Neeley’s fast implementation for triangular g ( XX + Y Y )coupling [23] ( cf. [24]) to arbitrary coupling g ( XX + Y Y )+ ˜ gZZ , including the Heisenberg exchange g = ˜ g : e − iα | W i = e + i ( π/ σ z U int e − i ( π/ σ y | i , with α = (5 g − g ) π/ g , t W = (4 / × ( π/ g ). V. CONCLUSION
In summary, we have developed several single-step symmetric implementations for generating maximally entangledtripartite quantum states in systems with anisotropic exchange interaction, which are directly applicable to supercon-ducting qubit architectures. In the GHZ case, both triangular and linear coupling schemes have been analyzed. Inthe isotropic limit, our implementations exhibit singularities that can be removed by breaking the symmetry of thelocal pulses.
Acknowledgments
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