Decoherence and gate performance of coupled solid-state qubits

Decoherence and gate performance of coupled solid-state qubits

Markus J. Storcz*and Frank K. Wilhelm

Sektion Physik and CeNS, Ludwig-Maximilians-Universita¨t, Theresienstrasse 37, 80333 Mu¨nchen, Germany 共Received 16 December 2002; published 23 April 2003; publisher error corrected 28 April 2003兲 Solid-state quantum bits are promising candidates for the realization of a scalable quantum computer.

However, they are usually strongly limited by decoherence due to the many extra degrees of freedom of a solid-state system. We investigate a system of two solid-state qubits that are coupled via␴z

(i)丢␴z

( j) type of coupling. This kind of setup is typical for pseudospin solid-state quantum bits such as charge or flux systems.

We evaluate decoherence properties and gate quality factors in the presence of a common and two uncorrelated baths coupling to␴z, respectively. We show that at low temperatures, uncorrelated baths do degrade the gate quality more severely. In particular, we show that in the case of a common bath, optimum gate performance of a controlled-PHASEgate can be reached at very low temperatures, because our type of coupling commutes with the coupling to the decoherence, which makes this type of coupling interesting as compared to previously studied proposals with ␴y

(i)丢␴y

( j) coupling. Although less pronounced, this advantage also applies to the controlled-NOTgate.

DOI: 10.1103/PhysRevA.67.042319 PACS number共s兲: 03.67.Lx, 03.65.Yz, 05.40.⫺a, 85.25.⫺j

I. INTRODUCTION

Quantum computation has been shown to perform certain tasks much faster than classical computers 关1–3兴. Presently, very mature physical realizations of this idea originate in atomic physics, optics, and nuclear magnetic resonance.

These systems are phase coherent in abundance, however, scaling up the existing few-qubit systems is not straightfor- ward. Solid-state quantum computers have the potential ad- vantage of being arbitrarily scalable to large systems of many qubits 关4 – 6兴. Their most important drawback is the coupling to the many degrees of freedom of a solid-state system. Even though recently, there has been fast progress in improving the decoherence properties of experimentally re- alized solid-state quantum bits关7–11兴, this remains a formi- dable task.

Quite a lot is known about decoherence properties of single solid-state qubits, see, e.g., Refs. 关12–14兴, but much less is known about systems of two or more coupled qubits 关15–17兴. However, only for systems of at least two qubits, the central issue of entanglement can be studied. The physi- cally available types of qubit coupling can be classified as Heisenberg-type exchange that is typical for real spin-1/2 systems, and Ising-type coupling, which is characteristic for pseudospin setups, where the computational degrees of free- dom are not real spins. In the latter, the different spin com- ponents typically correspond to distinct variables, such as charge and flux关10,18兴whose couplings can and have to be engineered on completely different footing. Previous work 关16,17兴presented the properties of a system of two coupled solid-state qubits that are coupled via ␴y

(i)丢␴y

( j) type cou- pling as proposed in Ref.关14兴as the current-current coupling of superconducting charge quantum bits.

On the other hand, many systems such as inductively coupled flux qubits 关6兴, capacitively coupled charge qubits 关7,8兴, and other pseudospin systems关19兴are described by a

␴z (i)丢␴z

( j) Ising-type coupling. This indicates that the com- putational basis states are coupled, which, i.e., in the case of flux qubits are magnetic fluxes, whereas ␴x/y are electric charges. The ␴z observable is a natural way of coupling, because it is typically easy to couple to. We will study a two qubit-system coupled this way that is exposed to Gaussian noise coupling to ␴z, the ‘‘natural’’ observable. This ex- ample accounts for the crucial effect of electromagnetic noise in superconducting qubits. We will compare both the cases of noise that affects both qubits in a correlated way and the case of uncorrelated single-qubit errors. We determine the decoherence properties of the system by application of the well-known Bloch-Redfield formalism and determine quality factors of a controlled-NOT 共^CNOT兲 gate for both types of errors and feasible parameters of the system.

II. MODEL HAMILTONIAN

We model the Hamiltonian of a system of two qubits, coupled via Ising-type coupling. Each of the two qubits is a two-state system that is described in pseudospin notation by the single-qubit Hamiltonian关13兴

H_sq⫽⫺1 2⑀␴^ˆz⫺1

2⌬␴^ˆx, 共1兲 where⑀is the energy bias and ⌬ the tunnel matrix element.

The coupling between the qubits is determined by an extra term in the Hamiltonian H_qq⫽⫺(K/2)␴^ˆz

(1)丢␴^ˆz

(2) that repre- sents e.g., inductive interaction 共directly or via flux trans- former兲in the case of flux qubits关6,20兴. Thus, the complete two-qubit Hamiltonian in the absence of a dissipative envi- ronment reads

H_2qb⫽i

兺

⫽1,2

冉

^{⫺12⑀i␴ˆz(i)⫺12⌬i␴ˆx(i)}

冊

^{⫺12K␴ˆz(1)␴ˆz(2). 共2兲}

The dissipative 共bosonic兲environment is conveniently mod- eled as either a common bath or two distinct baths of har-

*Electronic address: storcz@theorie.physik.uni-muenchen.de

or picked up by coupling elements such as flux transformers 关6兴. Short correlation length radiation or local readout and control electronics coupling to individual qubits 关13兴 might be described as coupling to two uncorrelated baths of har- monic oscillators.

One should note that if the number of qubits is increased to more than two, there might also occur dissipative effects that neither affect all qubits nor only a single qubit, but rather a cluster of qubits, thus, enhancing the complexity of our considerations 关21兴.

In the case of two uncorrelated baths, the full Hamiltonian reads

H_2qb^2b ⫽i⫽

兺

1,2

冉

^{⫺12⑀i␴ˆz(i)⫺12⌬i␴ˆx(i)⫹12␴ˆz(i)Xˆ(i)}

冊

⫺1 2K␴^ˆz

(1)␴^ˆz (2)⫹H_B

1⫹H_B

2, 共3兲

where each qubit couples to its own, distinct harmonic oscil- lator bath H_B

i, i⫽1,2, via the coupling term ␴^ˆz (i)Xˆ⁽ⁱ⁾, i

⫽1,2, that bilinearly couples a qubit to the collective bath coordinate Xˆ⁽ⁱ⁾⫽␨兺_␯␭_␯x_␯. We again sum over the two qu- bits. In the case of two qubits coupled to one common bath, we model our two-qubit system with the Hamiltonian

H_2qb^1b ⫽⫺1

2 _i⫽

兺

_1,2^{共⑀i␴ˆz(i)⫹⌬i␴ˆx(i)兲⫺1}2K␴^ˆz (1)␴^ˆz

(2)

⫹1 2共␴^ˆz

(1)⫹␴^ˆz

(2)兲Xˆ⫹H_B, 共4兲

where H_Bdenotes one common bath of harmonic oscillators.

The appropriate starting point for our further analysis is the singlet/triplet basis, consisting of 兩↑↑典ª(1,0,0,0)^T, (1/

冑

²⁾⁽兩↑↓典^⫹兩↓↑典⁾ª(0,1,0,0)^T, 兩↓↓典ª(0,0,1,0)^T, and the singlet state (1/

冑

²⁾⁽兩↑↓典⫺兩↓↑典⁾ª(0,0,0,1)^T. In the case of flux qubits, the ↑ and↓ states correspond to clockwise and counterclockwise currents respectively.

In this basis, the undamped Hamiltonian H_2qb, Eq.共2兲, of the two-qubit system assumes the matrix form

H_2qb⫽⫺1

2

冉

^{⫺⌬⑀⫹␩0K␩ ⫺⌬␩␩K⑀ K⌬␩⫺0␩⑀ ⫺⫺⌬⌬⌬␩K⑀␩}

冊

^{, 共5兲}

with ⑀⫽⑀1⫹⑀2, ␩⫽(⌬1⫹⌬2)/

冑

^2, ⌬␩⫽(⌬1⫺⌬2)/

冑

^2,

and ⌬⑀⫽⑀1⫺⑀2. From now on, for simplicity, we concen- trate on the case of equal parameter settings, ⌬1⫽⌬2 and

⑀1⫽⑀2.

H_2qb^2b ⫽⫺2

冉

^{00 ⫺␩⌬s K⫺0⑀⫹s ⫺0K}

冊

^{, 共6兲}

with s⫽X₁⫹X₂ and⌬s⫽X₁⫺X₂. Here, the bath mediates transitions between the singlet and triplet states, the singlet is not a protected subspace.

In the case of two qubits with equal parameters that are coupled to one common bath, we obtain the matrix

H_2qb^1b ⫽⫺1

2

冉

^{⑀⫺s␩00⫹K ⫺␩␩0K K⫺␩00⑀⫹s ⫺000K}

冊

^{, 共7兲}

where s⫽2X and⌬s⫽0. One directly recognizes that com- pared to Eq. 共6兲 in this case, thermalization to the singlet state is impeded, because Eq. 共7兲 is block diagonal in the singlet and triplet subspaces. The singlet and triplet are com- pletely decoupled from each other, and in the case of one common bath the singlet is also completely decoupled from the bath and thus, protected from dissipative effects. There- fore, a system in contact with one common bath that is pre- pared in the singlet state will never experience any decoher- ence effects. The singlet state is a decoherence free subspace 共DFS兲 关22兴, although a trivial, one-dimensional one.

III. EIGENENERGIES AND EIGENSTATES OF THE TWO-QUBIT HAMILTONIAN

We calculate exact analytical eigenvalues and eigenvec- tors of the unperturbed two-qubit system Hamiltonian in the aforementioned symmetric case of Eq.共5兲, which reads

H_2qb⫽⫺1

2

冉

^{⑀⫹␩00K ⫺␩␩0K K␩⫺00 ⑀ ⫺000K}

冊

^{. 共8兲}

This Hamiltonian is block diagonal and the largest block, the triplet, is three dimensional, i.e., it can be diagonalized using Cardano’s formula. Details of that calculation are given in Ref. 关23兴. The case of nonidentical qubits is more easily handled numerically.

In the following,兩E1典^,兩E2典^, 兩E3典^{, and}兩E4典 ^{denote the} eigenstates of the two-qubit system. The eigenenergies of the unperturbed Hamiltonian共8兲depend on the three parameters K,⑀^{, and}␩. Fig. 1 displays the eigenenergies in more detail for typical experimentally accessible values. The values that are chosen for the parameters ⑀^, ␩, and K in Fig. 1 corre- spond to what can be reached in flux qubits. They typically assume values of a few GHz resembling the parameters

of known single- and two-qubit experiments in Delft 关13兴 and at MIT关24兴. Therefore, we will use a characteristic en- ergy scale E_s, which is typically E_s⫽1 GHz. The corre- sponding scales are t_s⫽1 ns, ␻s⫽2␲⫻1 GHz, and T_s

⫽␯s(h/k_B)⫽4.8⫻10^⫺2 K. Panel 共1兲 shows that for large values of⑀, two of the eigenenergies are degenerate共namely, for ⑀Ⰷ␩,K the states 兩E1典 ^and 兩E4典 equal the states (1/

冑

²⁾⁽兩↑↓典⫺兩↓↑典^{) and (1/}

冑

²⁾⁽兩↑↓典⫹兩↓↑典), hence the eigenenergies are degenerate兲 while near zero energy bias 共magnetic frustation f⫽1/2) all four eigenenergies might be distinguished. Note also that, therefore, at zero energy bias, the transition frequency ␻14⫽⫺␻41 has a local maximum, which, as will be shown below, can only be accessed via nonsymmetric driving.

If K is set to a big positive value corresponding to large ferromagnetic coupling 关Fig. 1, panel 共2兲, K⫽10E_s], the Hamiltonian 共8兲 is nearly diagonal and, hence, the eigen- states in good approximation are equal to the singlet/triplet basis states. In this case, 兩E3典 equals the triplet state (1/

冑

²⁾⁽兩↑↓典⫹兩↓↑典^), 兩E2典 ^and兩E4典 ^equal 兩↑↑典 ^and兩↓↓典^, respectively, for positive values of⑀. For large negative val- ues of ⑀, the two states兩E2典 ^and兩E4典 become equal兩↓↓典 and 兩↑↑典 with a pseudo-spin-flip between clockwise and counterclockwise rotating currents at⑀⫽0 when going from positive to negative ⑀. In the case of large ferromagnetic coupling, the ground state tends towards the superposition (1/

冑

²⁾⁽兩↑↑典⫹兩↓↓典^{). Panel}共2兲shows that only for ⑀ ^equal to zero, both 兩E2典⫽兩↑↑典 ⁽兩E2典⫽兩↓↓典, for negative⑀^{) and} 兩E4典^⫽兩↓↓典⁽兩E4典^⫽兩↑↑典, for negative⑀) have the same en- ergies共which one would expect if the⫺(1/2)K␴z

(1)␴z (2)term in the Hamiltonian dominates兲, because if⑀is increased, the

⑀i␴^ˆz

(i) (i⫽1,2) terms in the Hamiltonian change the energy.

For large antiferromagnetic coupling, 兩⫺K兩Ⰷ⑀^,⌬ the states 兩↑↓典 ^and兩↓↑典 are favorable. In this limit, the ground state tends towards (1/

冑

²⁾⁽兩↑↓典⫹兩↓↑典) and the energy splitting between (1/

冑

²⁾⁽兩↑↓典⫹兩↓↑典^{) and (1/}

冑

²⁾⁽兩↑↓典

⫺兩↓↑典) vanishes asymptotically, leaving the ground state nearly degenerate.

From Fig. 1, panel 共3兲, one directly recognizes that the singlet eigenenergy crosses the triplet spectrum, which is a consequence of the fact that the singlet does not interact with any triplet states. At zero energy bias 共magnetic frustration f⫽1/2, for a flux qubit兲, none of the eigenstates equal one of the triplet basis states 共e.g., as observed for a large energy bias⑀), they are rather nontrivial superpositions. This is elu- cidated further in the following paragraph. The inset of panel 共2兲depicts the level anticrossing between the eigenenergies of the two states 兩E2典 ^and兩E4典 due to quantum tunneling.

In general, the eigenstates are a superposition of singlet/

triplet states. Figure 2 shows how singlet/triplet states com- bine into eigenstates for different qubit parameters. The first eigenstate 兩E1典 ^{equals (1/}

冑

²⁾⁽兩↑↓典⫺兩↓↑典) for all times while the other eigenstates兩E2典^,兩E3典^{, and兩E4}典 are in gen- eral superpositions of the singlet/triplet basis states. For large values of兩⑀兩, the eigenstates approach the singlet/triplet兲ba- sis states. In particular, at typical working points, where ⑀

⬇5⌬ 关13兴, the eigenstates already nearly equal the singlet/

triplet basis states. Hence, although the anticrossing de- scribed above corresponds to the anticrossing used in Refs.

关9,25兴to demonstrate Schro¨dinger’s cat states, entanglement is prevalent away from the degeneracy point. For an experi- mental proof, one still would have to show that one has successfully prepared coherent couplings by spectroscopi- cally tracing the energy spectrum. Note that, for clarity, in FIG. 1. Plot of the eigenenergies of the eigenstates兩E1典^, 兩E2典^,兩E3典^{, and}兩E4典. From upper left to lower right:共1兲K⫽␩⫽E_sand⑀is varied,共2兲K⫽10Es, ␩⫽Es, and⑀ is varied; the inset resolves the avoided level crossing due to the finite transmission amplitude␩^;共3兲

␩⫽⑀⫽E_sand K is varied;共4兲K⫽⑀⫽E_sand␩ ^{is varied.}

Fig. 2, the interqubit coupling strength K is fixed to a rather large value of E_sthat also sets the width of the anticrossing, which potentially can be very narrow.

Spectroscopy

As a first technological step towards demonstrating coher- ent manipulation of qubits, usually the transition frequencies between certain energy levels are probed关9,25兴, i.e., the en- ergy differences between the levels. Figures 3 and 4 depict the transition frequencies between the four eigenstates. The transition frequencies are defined as␻nm⫽(E_n⫺E_m)/ប and

␻nm⫽⫺␻mn. The transitions between the singlet state兩E1典 and the triplet states are forbidden in the case of one com- mon bath, due to the special symmetries of the Hamiltonian 共4兲, if the system is driven collectively through a time- dependent energy bias⑀1(t)⫽⑀2(t). However, in the case of two distinct baths, the environment can mediate transitions between the singlet and the triplet states.

Not all transition frequencies have local minima at ⑀

⫽0. The frequencies␻41and␻34have local maxima at zero energy bias⑀. This can already be inferred from Fig. 1, panel 共1兲, the energy of the eigenstate兩E4典has a local minimum at

⑀⫽0. Similarily, the substructure of␻34 can be understood from Fig. 1: the frequency ␻34 has a local maximum at ⑀ FIG. 2. Plot of the amplitude of the different singlet共triplet兲states of which the eigenstates denoted by兩E1典^, 兩E2典^, 兩E3典^{, and}兩E4典^are composed for the four eigenstates. In all plots⑀is varied, and K and␩are fixed to E_s.

FIG. 3. Plot of the absolute value of the transition frequencies

␻32, ␻42, and␻31. In the left column K⫽␩⫽0.2E_sand⑀ is var- ied. In the right column, K⫽0.2Es, ⑀⫽Es, and␩^{is varied.}

FIG. 4. Plot of the transition frequencies␻21,␻41, and␻34. In the left column, K⫽␩⫽0.2E_sand⑀is varied. In the right column, K⫽0.2Es, ⑀⫽Es, and␩^{is varied.}

⫽0, because of the local minimum of the eigenenergy of the state兩E4典. First, if⑀is increased, the level spacing of兩E4典 and 兩E3典 decreases. Then, for larger values of ⑀, the level spacing of兩E4典^and兩E3典increases again. Thus, the structure observed for ␻34around⑀⫽0 emerges in Fig. 4.

IV. BLOCH-REDFIELD FORMALISM

In order to describe decoherence in the weak damping limit, we use the Bloch-Redfield Formalism关26兴. It provides a systematic way of finding a set of coupled master equations which describes the dynamics of the reduced共i.e., the reser- voir coordinates are traced out兲 density matrix for a given system in contact with a dissipative environment and has recently been shown to be numerically equivalent to the more elaborate path-integral scheme 关27兴. The Hamiltonian of our two-qubit system in contact with a dissipative envi- ronment, Eqs. 共3兲 and 共4兲, has the generic ‘‘system⫹bath’’

form

H_{o p}共t兲⫽H_2qb⫹H_B⫹H_int, 共9兲 where H_B is a bath of harmonic oscillators and H_int inherits the coupling to a dissipative environment. In our case, the effects of driving are not investigated. In Born approxima- tion and when the system is only weakly coupled to the environment, Bloch-Redfield theory provides the following set of equations for the reduced density matrix␳ ^describing the dynamics of the system关28,29兴:

␳^˙nm共t兲⫽⫺i␻nm␳nm共t兲⫺

兺

_kl ^{Rnmkᐉ␳kᐉ共t兲, 共10兲}

where ␻nm⫽(E_n⫺E_m)/ប, and max_n,m,k,ᐉ兩Re(R_nmkᐉ)兩

⬍min_n⫽m兩␻nm兩 must hold. The Redfield relaxation tensor R_nmkᐉcomprises the dissipative effects of the coupling of the system to the environment. The elements of the Redfield relaxation tensor are given through golden rule rates关28兴

R_nmkᐉ⫽␦_ᐉm

兺

_r ^{⌫nrrk(⫹)⫹␦nk}

兺

_r ^{⌫ᐉ(⫺rrm)}

⫺⌫_ᐉmnk (⫹) ⫺⌫_ᐉmnk

(⫺)

. 共11兲

A. Two qubits coupled to two distinct baths

We now evaluate the Golden rule expressions in Eq.共11兲 in the case of two qubits, each coupled to a distinct harmonic oscillator bath. Here, H˜

I(t)⫽exp(iH_Bt/ប)H_Iexp(⫺iH_Bt/ប) de- notes the coupling between system and bath in the interac- tion picture, and the bracket denotes thermal average of the bath degrees of freedom. Writing down all contributions gives

⌫_ᐉmnk

(⫹) ⫽ប^⫺2

冕

0

⬁

dte^⫺i␻nkt具^{e[i(HB1⫹HB2)t/ប]}

⫻共␴z,ᐉm

(1) 丢Xˆ⁽¹⁾⫹␴z,ᐉm

(2) 丢Xˆ⁽²⁾兲e^[⫺i(HB1⫹H_B

2)t/ប]

⫻共␴z,nk

(1) 丢Xˆ⁽¹⁾⫹␴z,nk

(2) 丢Xˆ⁽²⁾兲典^{, 共12兲}

where ␴z,nm

(i) (i⫽1,2) are the matrix elements of ␴^ˆz (i) with respect to the eigenbasis of the unperturbed Hamiltonian共8兲 and likewise for⌫_ᐉmnk

(⫺)

.

We assume Ohmic spectral densities with a Drude cutoff.

This is a realistic assumption, i.e., for electromagnetic noise 关13兴and leads to integrals in the rates which are tractable by the residue theorem. The cutoff frequency␻cfor the spectral functions of the two qubits is typically assumed to be the largest frequency in the problem, this is discussed further in Sec. IV E,

J₁共␻兲⫽ ␣1ប␻ 1⫹␻²

␻c 2

and J₂共␻兲⫽ ␣2ប␻ 1⫹␻²

␻c 2

. 共13兲

The dimensionless parameter␣ describes the strength of the dissipative effects that enter the Hamiltonian via the coupling to the environment, described by s and⌬s. In order for the Bloch-Redfield formalism, which involves a Born approxi- mation in the system-bath coupling, to be valid, we have to assume ␣1/2Ⰶ1. After tracing out over the bath degrees of freedom, the rates read

⌫_ᐉmnk (⫹) ⫽ 1

8ប 关⌳¹J₁共␻nk兲⫹⌳²J₂共␻nk兲兴关coth共␤ប␻nk/2兲⫺1兴

⫹ i

4␲ប 关⌳²M^⫺共␻nk,2兲⫹⌳¹M^⫺共␻nk,1兲兴 共14兲 with⌳¹⫽⌳_ᐉmnk

1 ⫽␴z,ᐉm (1) ␴z,nk

(1) ,⌳²⫽⌳_ᐉmnk 2 ⫽␴z,ᐉm

(2) ␴z,nk (2) , and

M^⫾共⍀,i兲⫽P

冕

^0⬁d␻_␻^J2i_⫺^共␻_⍀^{兲2关coth共␤ប␻/2兲⍀⫾␻兴,}

共15兲 hereP denotes the principal value. Likewise,

⌫_ᐉmnk (⫺) ⫽ 1

8ប 关⌳¹J₁共␻_ᐉm兲⫹⌳²J₂共␻_ᐉm兲兴关coth共␤ប␻_ᐉm/2兲⫹1兴

⫹ i

4␲ប 关⌳²M^⫹共␻ᐉm,2兲⫹⌳¹M^⫹共␻ᐉm,1兲兴. 共16兲 The rates⌫_ᐉmnk

(⫹) and⌫_ᐉmnk

(⫺) might be inserted into Eq.共11兲to build the Redfield tensor. Note, here, that for ␻nk→0, and

␻ᐉm→0 respectively, the real part of the rates共which is re- sponsible for relaxation and dephasing兲 is of value ⌫_ᐉmnk (⫹)

⫽⌫_ᐉmnk

(⫺) ⫽(1/4␤ប)关␴z,ᐉm (1) ␴z,nk

(1) ␣1⫹␴z,lm (2) ␴z,nk

(2)␣2兴.

To solve the set of differential equations共10兲, it is conve- nient to collapse ␳ into a vector. In general, the Redfield equations共10兲without driving are solved by an ansatz of the type ␳^(t)⫽Bexp(R˜ t)B^⫺1␳(0), where R˜ is a diagonal matrix.

The entries of this diagonal matrix are the eigenvalues of the Redfield tensor 共11兲, written in matrix form, including the dominating term i␻nm关cf. Eq.共10兲兴. Here, the reduced den- sity matrix ␳⫽(␳11, . . . ,␳44)^T is written as a vector. The

matrix B describes the basis change to the eigenbasis of R˜ , in which R˜ has diagonal form.

B. Two qubits coupled to one common bath

For the case of two qubits coupled to one common bath, we perform the same calculation as in the preceding section, which leads to expressions for the rates analogous to Eqs.

共16兲

⌫_ᐉmnk (⫹) ⫽ 1

8ប ⌳J共␻nk兲关coth共␤ប␻nk/2兲⫺1兴⫹ i⌳ 4␲ប

⫻P

冕

^0⬁d␻_␻^J2_⫺^共␻_␻^兲_nk^{2 关coth共␤ប␻/2兲␻nk⫺␻兴,}

共17兲 with ⌳⫽⌳_ᐉmnk⫽␴z,ᐉm

(1) ␴z,nk (1) ⫹␴z,ᐉm

(1) ␴z,nk (2)⫹␴z,ᐉm

(2) ␴z,nk (1)

⫹␴z,ᐉm (2) ␴z,nk

(2) and

⌫_ᐉmnk (⫺) ⫽ 1

8ប ⌳J共␻ᐉm兲关coth共␤ប␻ᐉm/2兲⫹1兴⫹ i⌳ 4␲ប

⫻P

冕

0

⬁

d␻ ^J共␻兲

␻²⫺␻_ᐉm

2 关coth共␤ប␻^/2兲␻_ᐉm⫹␻兴. 共18兲 The difference between the rates for the case of two distinct baths 共14兲and 共16兲 are the two extra terms ␴z,ᐉm

(1) ␴z,nk (2) and

␴z,ᐉm (2) ␴z,nk

(1) . They originate when tracing out the bath degrees of freedom. In the case of one common bath, there is only one spectral function, which we also assume to be Ohmic J(␻⁾⫽(␣ប␻^)/(1⫹␻^2/␻c

2). For ␻nk→0, and␻_ᐉm→0, re- spectively, the real part of the rates is of the value ⌫_ᐉmnk (⫹)

⫽⌫_ᐉmnk

(⫺) ⫽(␣^/4␤ប)⌳, for ␻ᐉm,␻nk→0.

C. Dynamics of coupled flux qubits with dissipation The dissipative effects affecting the two-qubit system lead to decoherence, which manifests itself in two ways. The sys- tem experiences energy relaxation on a time scale ␶R⫽⌫R⫺1

(⌫R is the sum of the relaxation rates of the four diagonal elements of the reduced density matrix; ⌫R⫽⫺兺n⌰n and

⌰n are the eigenvalues of the matrix that consists of the tensor elements R_n,m,n,m, n,m⫽1, . . . ,4兲, called relaxation time, into a thermal mixture of the system’s energy eigen- states. Therefore, the diagonal elements of the reduced den- sity matrix decay to the value given by the Boltzmann factors. The quantum coherent dynamics of the system are superimposed on the relaxation and decay on a usually shorter time scale␶_{␸i j}⫽⌫_␸^⫺_{i j}¹ (i, j⫽1, . . . ,4;i⫽j and

⌫_␸nm⫽⫺ReR_n,m,n,m^1b,2b ) termed dephasing time. Thus, dephas- ing causes the off-diagonal terms共coherences兲of the reduced density matrix to tend towards zero.

First, we investigate the incoherent relaxation of the two- qubit system out of an eigenstate. At long times, the system is expected to reach thermal equilibrium, ␳eq⫽(1/Z)e^⫺␤H. Special cases are T⫽0, where ␳eq equals the projector on FIG. 5. Plot of the occupation probability of the four eigenstates 兩E1典^, 兩E2典^, 兩E3典^{, and} 兩E4典 ^for initially starting in one of the eigenstates兩E1典 共first row兲,兩E2典 共second row兲, or兩E3典 共third row兲 at T⫽0. The left column illus- trates the case of two qubits cou- pling to one common bath and the right column the case of two qu- bits coupling to two distinct baths.

The energies K,⑀, and ␩ ^{are all} fixed to Es. The characteristic time scale t_sis t_s⫽1/␯s.

the ground state and T→⬁, where all eigenstates are occu- pied with the same probability, i.e., ␳eq⫽(1/4)1ˆ. Figures 5 and 6 illustrate the relaxation of the system prepared in one of the four eigenstates for temperatures T⫽0 and T

⫽21T_s respectively. The qubit energies K, ⑀^{, and} ␩ ^{are all} set to E_s and␣ ^{is set to}␣⫽10^⫺3. From Fig. 1, one recog- nizes relaxation into the eigenstate兩E2典, the ground state for this set of parameters.

At low temperatures (T⫽0), we observe that for the case of two distinct uncorrelated baths, a system prepared in one of the four eigenstates always relaxes into the ground state.

In the case of two qubits coupling to one common bath, this is not always the case, as can be seen in the upper left panels of Figs. 5 and 6. This can be explained through our previous observation, that the singlet is a protected subspace: Neither the free nor, unlike in the case of distinct baths, the bath- mediated dynamics couple the singlet to the triplet space.

Moreover, we can observe that relaxation to the ground state happens by populating intermediate eigenstates with a lower energy than the initial state the system was prepared in at t

⫽0 共cf. Fig. 1兲.

For high temperatures (T⯝21T_s), the system thermalizes into thermal equilibrium, where all eigenstates have equal occupation probabilities. Again, in the case of one common bath, thermalization of the singlet state is impeded and the three eigenstates兩E2典^,兩E3典^and兩E4典 have equal occupation probabilites of 1/3 after the relaxation time.

If the system is prepared in a superposition of eigenstates, e.g.,兩E3典 ^and兩E4典 as in Fig. 7, which are not in a protected subspace, we observe coherent oscillations between the eigenstates that are damped due to dephasing and after the decoherence time, the occupation probability of the eigen- states is given by the Boltzmann factors. This behavior is depicted in Fig. 7. Here, for␣⫽10^⫺3, the cases of T⫽0 and T⫽2.1T_s are compared. When the temperature is low enough, the system will relax into the ground state兩E2典^{, as} illustrated by the right column of Fig. 7. Thus, the occupa- tion probability of the state (1/

冑

²⁾⁽兩E3典^⫹兩E4典^{) goes to} zero. Here, in the case of zero temperature, the decoherence

times for the case of one common or two distinct baths are of the same order of magnitude. The left column illustrates the behavior when the temperature is increased. At T⫽2.1T_s, the system relaxes into an equally populated state on times much shorter than for T⫽0. For low temperatures, the char- acteristic time scale for dephasing and relaxation is some- what shorter for the case of one common bath (␶^1b/␶^2b

⬇0.9, for␣⫽10^⫺3). This can be explained by observing the temperature dependence of the rates shown in Fig. 8. Though for the case of one common bath, two of the dephasing rates are zero at T⫽0, the remaining rates are always slightly bigger for the case of one common bath compared to the case of two distinct baths. If the system is prepared in a general superposition, here 兩E3典 ^and兩E4典, nearly all rates become important thus compensating the effect of the two rates that are approximately zero at zero temperature and leading to faster decoherence.

If ␣ and, therefore, the strength of the dissipative effects is increased from␣⫽10^⫺3 to␣⫽10^⫺2, the observed coher- ent motion is significantly damped. Variation of␣ ^{leads to a} phase shift of the coherent oscillations, due to renormaliza- tion of the frequencies关16兴. However, in our case, the effects of renormalization are very small, as discussed in Sec. IV E, and cannot be observed in our plots.

D. Temperature dependence of the rates

Figure 8 displays the dependence of typical dephasing rates and the relaxation rate⌫R on temperature. These deco- herence rates are the inverse decoherence times. The rates are of the same magnitude for the cases of one common bath and two distinct baths. As a notable exception, in the case of one common bath, the dephasing rates⌫_␸21⫽⌫_␸12go to zero when the temperature is decreased, while all other rates satu- rate for T→0. This phenomenon is explained later on. If the temperature is increased from T_s⫽(h/k_B)␯s⫽4.8⫻10^⫺2 K, the increase of the dephasing and relaxation rates follows a power-law dependence. It is linear in temperature T with a slope given by the prefactors of the expression in the Red- FIG. 6. Plot of the occupation probability of the four eigenstates 兩E1典^{, 兩E2}典^{, 兩E3}典^{, and 兩E4}典 ^for initially starting in one of the eigenstates 兩E1典 共upper row兲 or 兩E2典 共lower row兲 at T⫽21T_s. The left column illustrates the case of two qubits coupling to one common bath and the right col- umn the case of two qubits cou- pling to two distinct baths. The energies K,⑀, and␩are all fixed to E_s. The characteristic time scale tsis ts⫽1/␯s.

FIG. 7. Plot of the occupation probability P_{(1/冑2)(兩E3典⫹兩E4典)}(t) when starting in the initial state (1/冑²⁾⁽兩E3典⫹兩E4典), which is a super- position of eigenstates兩E3典^and兩E4典. The first row shows the behavior for two qubits coupling to two uncorrelated baths. The lower row shows the behavior for two qubits coupled to one common bath. The qubit parameters ⑀, ␩, and K are set to E_s and ␣ is set to ␣

⫽10^⫺3. The inset resolves the time scale of the coherent oscillations.

FIG. 8. Log-log plot of the temperature dependence of the sum of the four relaxation rates and selected dephasing rates. Qu- bit parameters K,⑀, and␩^{are all} set to E_sand␣⫽10^⫺3. The upper panel shows the case of one com- mon bath and the lower panel the case of two distinct baths. At the characteristic temperature of ap- proximately 0.1T_s, the rates in- crease very steeply.

field rates that depends on temperature. At temperature T

⬇0.1T_s, the rates show a sharp increase for both cases. This roll-off point is set by the characteristic energy scale of the problem, which in turn is set by the energy bias⑀, the trans- mission matrix element␩, and the coupling strength K. For the choice of parameters in Fig. 8, the characteristic energy scale expressed in temperature is T⬇0.1T_s.

Note that there is also dephasing between the singlet and the triplet states. When the system is prepared 共by applica- tion of a suitable interaction兲 in a coherent superposition of singlet and triplet states, the phase evolves coherently. Then two possible decoherence mechanisms can destroy phase co- herence. First, ‘‘flipless’’ dephasing processes, where具^E典 ^re- mains unchanged. These flipless dephasing processes are de- scribed by the terms for␻ᐉm,␻nk→0 in the rates, Eqs.共16兲 and 共18兲. Obviously, these terms vanish for T→0, as the low-frequency component of Ohmic Gaussian noise is strictly thermal. Second, relaxation due to emission of a bo- son to the bath is also accompanied by a loss of phase co- herence. This process in general has a finite rate at T⫽0.

This explains the T dependence of the rates in the single-bath case:兩E1典alone is protected from the environment. As there are incoherent transitions between the triplet eigenstates even at T⫽0, the relative phase of a coherent oscillation betweeen 兩E1典 and any of those is randomized, and the decoherence rates ⌫_␸3/4,1 are finite even at T⫽0. As a notable exception, 兩E2典, the lowest-energy state in the triplet subspace, can only be flipped through absorption of energy, which implies that the dephasing rate⌫_␸21also vanishes at low temperature.

The described behavior can be observed in Fig. 8.

If the parameters⑀ ^and␩ are tuned to zero, thus K being the only nonvanishing parameter in the Hamiltonian, all dephasing and relaxation rates will vanish for T⫽0 in the

case of one common bath. This behavior is depicted in Fig.

9. It originates from the special symmetries of the Hamil- tonian in this case and the fact that for this particular two- qubit operation, the system Hamiltonian and the coupling to the bath are diagonal in the same basis. This special case is of crucial importance for the quantum gate operation as de- scribed in Sec. V and affects the gate quality factors.

E. Renormalization effects

Next to causing decoherence, the interaction with the bath also renormalizes the qubit frequencies. This is mostly due to the fast bath modes, and can be understood analogous to the Franck-Condon effect, the Lamb shift, or the adiabatic renor- malization 关30兴. Renormalization of the oscillation frequen- cies ␻nmis controlled by the imaginary part of the Redfield tensor关16兴

␻nm→␻^˜nmª␻nm⫺ImR_nmnm. 共19兲

Note that ImR_nmnm⫽⫺ImR_mnmndue to the fact that the cor- relators in the Golden Rule expressions have the same parity.

The imaginary part of the Redfield tensor is given by

Im⌫_ᐉmnk

(⫹) ⫽C_ᐉ^1b,2b_mnk 1

␲បP

冕

^{0⬁d␻J共␻兲}

冉

^{␻2⫺1␻nk2}

冊

⫻关coth共␤ប␻^/2兲␻nk⫺␻兴 共20兲 and

FIG. 9. Plot of the temperature dependence of the sum of the four relaxation rates and selected dephasing rates. Qubit parameters

⑀ and ␩ are set to 0, K is set to E_s, and ␣⫽10^⫺3 corresponding to the choice of parameters used for the U_XORoperation. The upper panel shows the case of one com- mon bath and the lower panel the case of two distinct baths. In the case of one common bath the sys- tem will experience no dissipative effects at T⫽0.

Im⌫_ᐉmnk

(⫺) ⫽C_ᐉ^1b,2b_mnk 1

␲បP

冕

0

⬁

d␻^J共␻兲

冉

^␻2⫺1␻ᐉm 2

冊

⫻关coth共␤ប␻^/2兲␻_ᐉm⫹␻兴, 共21兲 whereP denotes the principal value, and C_ᐉ^1b,2b_mnk are prefac- tors defined, in the case of two distinct baths, according to C_ᐉ^2b_mnk⫽¹4关␴z,ᐉm

(1) ␴z,nk (1) ⫹␴z,ᐉm

(2) ␴z,nk

(2) 兴 and in the case of one common bath C_ᐉ^1b_mnk⫽¹4⌳. Here, for simplicity, we assumed

␣1⫽␣2⫽␣ and thus, J₁(␻⁾⫽J₂(␻⁾⫽J(␻). Evaluation of the integral leads to the following expression for ⌫_ᐉmnk

(⫹)

:

Im⌫_ᐉmnk

(⫹) ⫽C_ᐉ^1b,2b_mnk ␣␻c 2␻nk

2␲共␻c 2⫹␻nk

2 兲

冋

^{␺共1⫹c2兲⫹␺共c2兲}

⫺2Re关␺共ic₁兲兴⫺␲ ␻c

␻nk

册

^{, 共22兲}

with c₁ª(␤ប␻nk)/(2␲^{) and c}2ª(␤ប␻c)/(2␲^{). In the} case of ⌫_ᐉmnk

(⫺)

, the expression is

Im⌫_ᐉmnk

(⫺) ⫽C_ᐉ^1b,2b_mnk ␣␻c 2␻_ᐉm

2␲共␻c 2⫹␻_ᐉm

2 兲

冋

^{␺共1⫹c2兲⫹␺共c2兲}

⫺2Re关␺共ic₁兲兴⫹␲ ␻c

␻_ᐉm

册

^{, 共23兲}

with c₁ª(␻_ᐉm␤ប)/(2␲). The terms in Eqs.共22兲 and 共23兲 which are linear in␻cgive no net contribution to the imagi- nary part of the Redfield tensor关16兴. To illustrate the size of the renormalization effects, the ratio of the renormalization effects to the frequencies which are renormalized is depicted in Fig. 10.

If c₁ and c₂ are large, and the digamma functions can be approximated by a logarithm, the resulting expression for the renormalization effects will be independent of temperature.

The temperature dependence of Eqs. 共22兲and共23兲at higher temperatures, where c₁ and c₂ are small and the renormal- ization effects are very weak, is shown in Fig. 10. The rates 共22兲 and共23兲 diverge logarithmically with␻c in analogy to the well-known ultraviolet-divergence of the spin boson model 关30兴. When comparing the left (T⫽0) and right (T

⫽2.1T_s) panel, one recognizes that for the first case, one common bath gives somewhat smaller renormalization ef- fects than two distinct baths, while in the second case for T

⫽2.1T_s, the renormalization effects deviate only slightly 共see the behavior for␻23) and the renormalization effects are smaller for the case of two distinct baths. The effects of renormalization are always very small 关兩Im(R_n,m,n,m)/␻nm兩 below 1% for our choice of parameters兴 and are therefore, neglected in our calculations. However, having calculated Eqs. 共22兲 and共23兲, these are easily incorporated in our nu- merical calculations. The case of large renormalization ef- fects is discussed in Ref.关31兴.

We only plotted the size of the renormalization effects for

␻12, ␻14, and␻34, because in general, all values of␻nkare of the same magnitude and give similar plots. The size of the renormalization effects diverges linearly with ␣, the dimen- sionless parameter that describes the strength of the dissipa- tive effects.

For flux qubits, the cutoff frequency ␻c is given by the circuit properties. For a typical first order low-pass LR filter 关32兴in a qubit circuit 关13兴, one can insert R⫽50⍀ 共typical impedance of coaxial cables兲and L⬇1 nH共depends on the length of the circuit lines兲 into ␻LR⫽R/L, and gets that

␻LR⬇5⫻10¹⁰Hz. ␻LR is the largest frequency in the prob- lem 共see again Ref. 关13兴, Chap. 4.5兲 and ␻cⰇ␻LR should hold. Then ␻c⬇10¹³ Hz (⫽10⁴E_s) as cutoff frequency is a reasonable assumption.

V. GATE QUALITY FACTORS

In Sec. IV, we evaluated the dephasing and relaxation rates of the two-qubit system that is affected by a dissipative FIG. 10. The left plot depicts the ratio of the renormalization effects and the corresponding transition frequencies. Parameters: ␣

⫽10^⫺3, T⫽0, and␻c/␻sis varied between 10²and 10⁵for several frequencies (␻12, ␻14, and␻23) for the case of two baths and in the case of␻23also for the case of one common bath. The parameters for the right plot are␣⫽10^⫺3, T⫽2.1T_s, and␻c/␻sis varied between 10³and 10⁵. The inset of the left plot shows a log-log plot of the temperature dependence of the renormalization effects. Here␣⫽10^⫺3and

␻c⫽10¹³. Note that for small temperatures the renormalization effects do not depend on temperature.共This is elucidated further in Sec.

IV E.兲The plots are scaled logarithmically to emphasize the logarithmic divergence of the renormalization effects with␻c.

environment. Furthermore, we visualized the dynamics of the two-qubit system. This does not yet allow a full assess- ment of the performance as a quantum logic element. These should perform unitary gate operations and based on the rates alone, one can not judge how well quantum gate opera- tions might be performed with the two-qubit system. There- fore, to get a quantitative measure of how our setup behaves when performing a quantum logic gate operation, one can evaluate gate quality factors关33兴. The performance of a two- qubit gate is characterized by four quantities: the fidelity, purity, quantum degree, and entanglement capability. The fi- delity is defined as

F⫽ 1 16

兺

j⫽1

16

具⌿in j 兩U_G^⫹␳G

jU_G兩⌿in

j 典^, 共24兲

where U_G is the unitary matrix describing the desired ideal gate and the density matrix obtained from attempting a quan- tum gate operation in a hostile environment is ␳G

j⫽␳^(tG), which is evaluated for all initial conditions ␳⁽⁰⁾

⫽兩⌿in j 典具⌿in

j 兩. The fidelity is a measure of how well a quan- tum logic operation was performed. Without dissipation, the reduced density matrix␳G

j after performing the quantum gate operation, applying U_G and the inverse U_G^⫹ would equal

␳(0). Therefore, the fidelity for the ideal quantum gate op- eration should be 1.

The second quantifier is the purity

P⫽ 1

16_j

兺

_⫽¹⁶₁ ^{tr„共␳Gj兲2…, 共25兲}

which should be 1 without dissipation and 1/4 in a fully mixed state. The purity characterizes the effects of decoher- ence.

The third quantifier, the quantum degree, is defined as the maximum overlap of the resulting density matrix after the quantum gate operation with the maximally entangled states, the Bell states

Q⫽max

j,k 具^⌿me k 兩␳G

j兩⌿me

k 典^{, 共26兲}

where the Bell states⌿me

k are defined according to

兩⌿me

00⫽兩↓↓典⫹兩↑↑典

冑

^{2 , 兩⌿me01⫽}

兩↓↑典⫹兩↑↓典

冑

^{2 , 共27兲}

兩⌿me

10⫽兩↓↓典^⫺兩↑↑典

冑

^{2 , 兩⌿me11⫽}

兩↓↑典^⫺兩↑↓典

冑

^{2 . 共28兲}

For an ideal entangling operation, e.g., the controlled-NOT

gate, the quantum degree should be one. The quantum degree characterizes nonlocality. It has been shown关34兴that all den- sity operators that have an overlap with a maximally en- tangled state that is larger than the value 0.78关17兴violate the Clauser-Horne-Shimony-Holt inequality and are thus nonlo- cal.

The fourth quantifier, the entanglement capabilityC, is the smallest eigenvalue of the partially transposed density matrix for all possible unentangled input states兩⌿in

j 典^.共see below兲. It has been shown关35兴to be negative for an entangled state.

This quantifier should be⫺0.5, e.g., for the ideal U_XOR, thus characterizing a maximally entangled final state. Two of the gate quality factors, namely, the fidelity and purity might also be calculated for single-qubit gates 关12兴. However, en- tanglement can only be observed in a system of at least two qubits. Therefore, the quantum degree and entanglement ca- pability cannot be evaluated for single-qubit gates.

To form all possible initial density matrices, needed to calculate the gate quality factors, we use the 16 unentangled product states兩⌿in

j 典^{, j}⫽1, . . . ,16 defined关17兴according to 兩⌿a典1兩⌿b典2 (a,b⫽1, . . . ,4), with 兩⌿1典⫽兩↓典^, 兩⌿2典⫽兩↑典^, 兩⌿3典^⫽(1/

冑

²⁾⁽兩↓典^⫹兩↑典^{), and 兩⌿}4典^⫽(1/

冑

²⁾⁽兩↓典^⫹i兩↑典^).

They form one possible basis set for the superoperator ␯G

with ␳^(tG)⫽␯G␳⁽⁰⁾ 关17,33兴. The states are chosen to be unentangled for being compatible with the definition ofC.

A. Implementation of two-qubit operations 1. Controlled phase-shift gate

To perform the controlled-NOToperation, it is necessary to be able to apply the controlled phase-shift operation to- gether with arbitrary single-qubit gates. In the computational basis (兩00典^,兩01典^,兩10典^,兩11典), the controlled phase-shift opera- tion is given by

U_CZ共␸兲⫽

冉

^{1000 0100 0010 e000i␸}

冊

^{, 共29兲}

and for␸⫽␲, up to a global phase factor,

U_CZ⫽exp

冉

^i␲4␴z(1)

冊

^exp

冉

^i␲4␴z(2)

冊

^exp

冉

^{i␲4␴z(1)␴z(2)}

冊

^.

共30兲 Note that in Eq. 共30兲 only ␴z operations, which commute with the coupling to the bath, are needed. The controlled phase-shift operation together with two Hadamard gates and a single-qubit phase-shift operation then gives the controlled-

NOTgate.

2. Controlled-NOTgate

Due to the fact that the set consisting of the U_XOR 共or controlled-NOT兲gate and the one-qubit rotations, is complete for quantum computation 关36兴, the U_XOR gate is a highly important two-qubit gate operation. Therefore we further in- vestigate the behavior of the four gate quality factors in this case. The U_XORoperation switches the second bit, depending on the value of the first bit of a two bit system. In the com- putational basis, this operation has the following matrix form: