Arbitrary qudit gates by adiabatic passage - Physical Review Link

PHYSICAL REVIEW A 87, 032328 (2013)

Arbitrary qudit gates by adiabatic passage B. Rousseaux,1 S. Guérin,1,* and N. V. Vitanov2 1

Laboratoire Interdisciplinaire Carnot de Bourgogne, CNRS UMR 6303, Université de Bourgogne, BP 47870, 21078 Dijon, France 2 Department of Physics, Sofia University, 5 James Bourchier Boulevard, 1164 Sofia, Bulgaria (Received 25 October 2012; published 25 March 2013) We derive an adiabatic technique that implements the most general SU(d) transformation in a quantum system of d degenerate states, featuring a qudit. This technique is based on the factorization of the SU(d) transformation into d generalized quantum Householder reflections, each of which is implemented by a two-shot stimulated Raman adiabatic passage with appropriate static phases. The energy of the lasers needed to synthesize a single Householder reflection is shown to be remarkably constant as a function of d. This technique is directly applicable to a linear trapped ion system with d + 1 ions. We implement the quantum Fourier transform numerically in a qudit with d = 4 (defined as a quartit) as an example. DOI: 10.1103/PhysRevA.87.032328

PACS number(s): 03.67.Ac, 03.65.Aa, 03.67.Lx, 32.80.Qk

I. INTRODUCTION

Quantum information processing makes use largely of qubits implemented by two-state quantum systems since the associated mathematical structure is well understood and offers a simple, intuitive, and universal construction of quantum gates through the quantum circuit architecture [1]. However, the structure of quantum mechanics allows a priori a computation on more than two states at once, i.e., on the so-called qudits (featured by a d-state system), promising a reduction in the number of operations to implement algorithms and in the number of physical carriers of information (e.g., ions or atoms), and ultimately a direct computation without a sequence of gates [2]. The use of photonic qutrits has already shown important new features (see, for instance, Ref. [3]). The principles of quantum logic on qudits, similar to the circuit model of the qubit, have been investigated and have been shown to be difficult to implement in the existing proposals [4]. The synthesis of general SU(d) transformations of d-state systems is of primary importance. In particular, quantum information processing leans heavily on the quantum Fourier transform [1]. It can be achieved by the use of sequences of SU(2) operations (corresponding to transformations acting upon only two of the d states at once), which requires d 2 operations [4,5]. It has been shown recently that a general SU(d) transformation can be implemented much more efficiently, in only d interaction steps, by using generalized Householder reflections (HRs) Md (χ ; ϕ) = 1l + (eiϕ − 1)|χ χ | of dimension d [6]. This has been very recently investigated for the qutrit [7]. An HR transformation can remarkably result from the propagator in the degenerate manifold of a d-pod quantum system, which consists of d degenerate states coupled to a single common upper state by d simultaneous pulsed fields of well-defined detuning and areas [8]. This applies in particular in an ensemble of trapped ions in a linear Paul trap [9]. The quantum search Grover algorithm [10] can be, in principle, implemented without gates and circuits by such HR transformations [11].

*

[email protected]

1050-2947/2013/87(3)/032328(4)

On the other hand, concrete implementations of quantum information processing, and more generally of the control of quantum systems, suffers from the imperfection of the systems, of the lack of knowledge on the system, and of decoherence by the environment. In particular, the laser scheme proposed in Refs. [6,8] for the implementation of an HR transformation requires stringent conditions of detuning and pulse areas which makes it difficult to implement in practice. It also leads to a significant population transfer in the lossy excited state. Thus robust techniques that, in addition, restrict the dynamics to decoherence-free subspaces, are desirable. Adiabatic passage techniques, and especially the ones that make use of dark states, such as the stimulated Raman adiabatic passage (STIRAP), are designed to overcome these issues [12]. STIRAP has been shown, for instance, to allow the transfer between arbitrary initial and final superpositions of states [13]. The extension of such results to the construction of a general gate is a much more difficult task since it requires to implement in a controlled way the propagator that produces an arbitrary SU(d) transformation. This has been solved only by adiabatic techniques to generate an arbitrary single-qubit gate [14], specific two-qubit gates [15,16], and a continuous version of the Grover search algorithm [17]. In this paper, we solve the above issue by proposing a controlled adiabatic scheme that implements an arbitrary SU(d) transformation. More precisely, we show how to generate any HR by a two-shot STIRAP. Such processes are then sequentially composed according to the factorization of a SU(d) transformation into HRs. We show the remarkable property that the energy of the lasers is constant as a function of d to achieve the adiabatic synthesis. This implies a growth of the energy only as d for the implementation of an arbitrary SU(d) transformation. As an example, we show a numerical synthesis of the quantum Fourier transform on a quartit (i.e., a qudit with d = 4). We finally present a discussion for its practical implementation. II. THE MODEL AND THE HR BY ADIABATIC PASSAGE

We consider the (d + 1)-pod system comprising d + 1 ground states, the d states of the qudit, and an ancillary state, coupled to a common excited state by d pump lasers for the d first transitions and a Stokes laser for the last one involving the ancillary state (see Fig. 1). Such a quantum

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©2013 American Physical Society

´ B. ROUSSEAUX, S. GUERIN, AND N. V. VITANOV

...

gg...g,1 1

P1

P2

...

Pd

d+1

d

Pd

P1

S


2

eg...g,0

ge...g,0

...

g

d

a

g...eg,0

gg...e,0

FIG. 1. (Color online) Left: Level scheme of the (d + 1)-pod featuring the d states of the qudit (labeled from |1 to |d) and the ancillary state |a. The pump fields {P1 ,P2 , . . . ,Pd } couple the states of the qudit to an excited state, and a Stokes field S the ancillary to the excited state. The states can be identified with multipartite states from d + 1 trapped two-state ions through zero- and one-phonon states and individually coupled by the pump and Stokes fields. respectively (right).

system can be implemented by d + 1 ions trapped in a linear Paul trap. The system is designed such that one laser pulse couples one ion using a vibrational red sideband transition through the Coulomb interaction between all ions. The ground state for all ions and one phonon in the trap represents a state |gg . . . g,1 that is then coupled to the ground-state basis {|eg . . . g,0,|ge . . . g,0, . . . ,|gg . . . eg,0,|gg . . . ge,0} by d red sideband pumps P1 ,P2 , . . . ,Pd and a Stokes field S (see Fig. 1). The symbols Pj and S denote the Rabi frequency corresponding to, respectively, the pump field j and the Stokes field with their respective transition moment. The initial condition is composed of any superposition of the first d ground states and the fields are designed such that the Stokes-pumps field sequence, where all the pumps share a common pulse-shape dependence (assuming that they are produced from a single source), drives all the population to the ancillary state |a by STIRAP. The HR transformation is achieved by a reversed STIRAP (pumps-Stokes field sequence) (see Fig. 2). The pump amplitudes are designed using the MorrisShore transformation in order to derive an effective system [18]. The initial Hamiltonian in the {|a,|gg . . . g,1,|1,|2, . . . ,|d} basis ⎤ ⎡ ⎤ ⎡ P1 0 S ∗ [0]1×d ⎢ .. ⎥ ⎢ † ⎥ (1) H =⎣ S 0 P ⎦ , P := ⎣ . ⎦ [0]d×1

P

[0]d×d

Pd

can be transformed by a time-independent transformation T into an effective system Hamiltonian reading in the basis {|a,|gg . . . g,1,|χ } (where we have omitted to write the d − 1 uncoupled states {|N C1 ,|N C2 , . . . ,|N Cd−1 } for ⎡

1 + (eiϕ − 1)|χ1 |2 ⎢ (eiϕ − 1)χ χ ∗ 1 2 ⎢ Ud (tf ,ti ) ⎢ . ⎢ .. ⎣ (eiϕ − 1)χ1 χd∗

P

Seiφ

t FIG. 2. (Color online) Pulse sequence with two consecutive STIRAPs generating the HR transformation Md (χ ; ϕ). Here the ensemble of the d pump fields P is schematically represented by a single line. The second STIRAP is reversed with respect to the first one and contains the additional phase ϕ for the Stokes pulse.

Pi S

...

Md(χ;φ) :

P

S

e

1

S

simplicity): ⎡ HMS

0 = ⎣S 0

S∗ 0 P0

⎤ 0 1l2 P0 ⎦ , T = [0]d×2 0

[0]2×d , Vd×d

(2)

with Vd×d ≡ V = [|χ ,|N C1 ,|N C2 , . . . ,|N Cd−1 ], a unitary transformation of dimension d with ⎡ ⎤ χ1 ≡ c1 c2 . . . cd−1 ⎢ χ ≡ s c . . . c eiφ2 ⎥ 1 2 d−1 ⎢ 2 ⎥ ⎥. (3) |χ = ⎢ .. ⎢ ⎥ ⎣ ⎦ . χd ≡ sd−1 eiφd Here, sj ≡ sin(θj /2) and cj ≡ cos(θj /2), where 0 θj < π is an angle parametrizing the pump amplitudes as P ≡ P0 (t)|χ . This means that the pump pulses form a vector whose components precisely correspond to the ones of the state |χ times a common (time-dependent and real) amplitude P0 (t), i.e., P† V = P0 (t)[1,0, . . . ,0]. To these pump pulses corresponds the root-mean-square (rms) pump Rabi frequency

d 2 P0 j =1 |χj |2 . The relative phases φj , j = 2, . . . ,d of the pump Pj are with respect to the phase of P1 (φ1 ≡ 0). The Stokes pulse S(t) = S0 cos(ωS t + φS ) (of frequency ωS and static phase φS ), is delayed with respect to the pump pulse. The first STIRAP sequence, Stokes (with φS = 0)-pumps, induces a complete population transfer from the state |χ to the ancillary state |a while the states |N Cj are unchanged with no additional phase as their corresponding energies are zero. The second (reversed) STIRAP sequence, pumps-Stokes (with φS = ϕ), allows a population return with the phase ϕ leading to the overall transfer |χ → eiϕ |χ . We remark that, since the dynamics follows in the adiabatic limit a dark state which does not involve the excited state |gg . . . g,1, it is in practice accomplished with negligible transient population in the excited vibrational state. The propagator Ud (tf ,ti ) from the initial ti and final tf times, expressed in the basis of the qudit, results in the adiabatic limit in Ud (tf ,ti ) V diag[eiϕ 1 · · · 1]V † which finally reads, as a central result of this paper,

(eiϕ − 1)χ1 χ2 1 + (eiϕ − 1)|χ2 |2 .. . iϕ (e − 1)χ2∗ χd∗ 032328-2

... ... .. . ...

⎤ (eiϕ − 1)χ1 χd (eiϕ − 1)χ2 χd ⎥ ⎥ ⎥ ≡ Md (χ ; ϕ). .. ⎥ ⎦ . iϕ 2 1 + (e − 1)|χd |

(4)

ARBITRARY QUDIT GATES BY ADIABATIC PASSAGE


This propagator synthesizes thus the HR Md (χ ; ϕ) by adiabatic passage. In practice, one has to determine the angles θj , j = 1, . . . ,d − 1, from the given parameters χj , j = 1, . . . ,d, that define the transformation (the phases φj are simply given by the phases of the components χj ). One achieves this program using the relations from the components 2 2 2 of |χ : cd−1 = |χ1 |2 + |χ2 |2 + · · · + |χd−1 |2 , cd−2 cd−1 = 2 2 2 2 2 2 |χ1 | + |χ2 | + · · · + |χd−2 | , . . . , c2 c3 · · · cd−1 = |χ1 |2 + 2 |χ2 |2 , c12 c22 c32 · · · cd−1 = |χ1 |2 . III. NUMERICAL IMPLEMENTATION OF THE QUANTUM FOURIER TRANSFORM

We show numerically the implementation of this transformation in the case of a quantum Fourier transform (QFT) on a quartit, ⎤ ⎡ 1 1 1 1 i −1 −i ⎥ 1⎢ ⎥ ⎢1 (5) Fˆ4 = ⎢ ⎥, 1 −1 ⎦ 2 ⎣ 1 −1 1 −i −1 i that makes use of two specific HRs [6], Fˆ4 = M4 (χ2 ; π )M4 (χ1 ; π/2), 1 1 |χ1 = √ [ 0 1 0 −1 ]T , |χ2 = [ −1 1 1 1 ]T . 2 2

(6a) (6b)

15

ΩT

|S|

|S|

P2

|S|

P , j = 2,...,4 j

|S|

Figure 3 shows the numerical implementation of the two consecutive HRs on the quartit using a 5-pod system (the quartit states and an ancillary state coupled to a common state |ggggg,1). The QFT is obtained at the end of the total process, as the infidelity ||U4 (tf ,ti ) − Fˆ4 || is very close to zero. The number of nonsimultaneous pulses to reach this transform for the quartit here is 8. For a qudit (d states), the generalized quantum gate requires in general d HRs, that is, 4d nonsimultaneous pulses. We study below, as a function of d, (i) the total fluence of the amplitude of the laser fields (that is, its time integrated pulse area), which to the total Rabi frequency is proportional fluence FE := |S(t)|dt + dj=1 |Pi (t)|dt, and (ii) the laser intensity fluence, corresponding to the laser energy (per unit area), proportional to FI the fluence of the square rms pump Rabi frequency and the square Stokes Rabi frequency FI = [|P0 (t)|2 + |S(t)|2 ]dt. From the effective Hamiltonian (2), we immediately conclude that the process does not depend on d. This means the remarkable property that the energy required to achieve by adiabatic passage a single HR operation does not depend on d. This result is shown numerically in Fig. 4, where we have determined, for random HR (constructed from a random normalized vector |χ and taking ϕ = π ), the peak amplitude of P0 (t) from which the infidelity ||Ud − Md (χ ,π )|| goes below the accuracy 10−4 . On the other hand, we anticipate the total Rabi frequency fluence √ FE to be bounded by a function growing as d for large d, since the adiabaticity criterium is given by the area T P0,peak (which has to be much larger than one), with P0,peak the peak amplitude of P0 (t) and taking S0,peak ∼ P0,peak . We have indeed

0 P4

70

P1

√ P0,peak T d

Populations

−15 0.65 |1〉 |4〉

50

|gg...g,1〉

|2〉

||U4−F4||

0 2.5

√1 FE π

Fluence

|3〉 |a〉

30 0 −15

−10

−5

0

5

10

2 π FI

15

t/T FIG. 3. (Color online) Numerical simulation of the quantum Fourier transform on a quartit in a 5-pod system with two HRs. Upper: The (dimensionless) Rabi frequencies associated with the pump and Stokes fields. The sequence of the pump and Stokes lasers allows the implementation of the two required HRs, M4 (χ1 ; π/2), where only the pumps P2 and P4 are on, and next M4 (χ2 ; π ), where all the pumps are 2 on [according to Eq. (6b)]. All pulses are of shape e−(t/T ) and the time delays between Stokes and pumps are all T /1.4. Middle: Dynamics of the 5-pod system during the process for the particular initial condition |ψ(ti ) = 12 (|1 + i|2 + |3 + |4). Lower: Infidelity as (dimensionless) distance ||U4 − Fˆ4 || := Tr{[U4 (t,ti ) − Fˆ4 ]† [U4 (t,ti ) − Fˆ4 ]} between the numerical propagator U4 (t,ti ) (projected in the space defining the quartit states) and the quartit QFT Fˆ4 .

10 2

5

10

15

20

Dimension d FIG. 4. (Color online) Normalized square Rabi frequency fluence (proportional to the energy per area unit) FI (green crosses, in units of 1/T ) and normalized Rabi frequency fluence (proportional to the laser field amplitude fluence) FE (crosses, dimensionless, fitted by the bent solid red line) required to synthesize numerically a random HR with an infidelity below 10−4 as a function of the dimension d. Note that, for simplicity, only the pump fields have been considered in the present calculation. The curve fitting FE (bent solid red line) and √ the normalized Rabi frequency fluences are bounded by P0,peak T d (see text).

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Speak + dj=1 Pi,peak = Speak + P0,peak dj=1 χi Speak + √ P0,peak d. This is confirmed in Fig. 4. IV. CONCLUSION

In conclusion, we have proposed an adiabatic passage technique to synthesize an HR and thus an arbitrary SU(d) transformation in a (d + 1)-pod system where d ground states form the qudit and the last one an ancillary state. The HR is generated by a two-shot STIRAP where the pump field amplitudes and phases can be explicitly determined from the given parameters of the HR. The robustness and high-fidelity issues can be addressed by extending the recent techniques aiming at optimizing adiabatic passage for state-to-state problems [19]. A practical implementation in an ion trap system necessitates taking into account the decoherence of the excited vibrational and ionic states. The present dark-state dynamics allows already a negligible transient population in the n = 1 vibrational state in the adiabatic limit. To solve the decoherence

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issue of the excited ionic state, one can, for instance, consider d ions with two ground states |g1 and |g2 , the second ground state |g2 playing the role of the excited state in the scheme described above. This necessitates for each ion a second field tuned around the center-of-mass transition |g2 ,0 ↔ |e,0 and an appropriate field sequence and/or detuned transitions to populate the excited ionic states in a negligible way. We can remark that the implementation of the d-dimension HR could be alternatively obtained in a d-ion system using a symmetric cyclic chirped double-pump pulse (without an ancillary state, or a Stokes pulse) such that the second pump is ϕ out of phase from the first one. In this case, the excited vibrational state is transiently populated. However, modern pulseshaping technology would allow an ultrafast transfer [19]. ACKNOWLEDGMENTS

We acknowledge support from the European Marie Curie Initial Training Network GA-ITN-214962-FASTQUAST, and from the Conseil Régional de Bourgogne.

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