arXiv:1603.06867v1 [quant-ph] 22 Mar 2016 · 2 W 0 = 1 R j = U T W j-1 f j = 6 E Tr[R j P j (E)] {P...

Pauli Decomposition over Commuting Subsets:Applications in Gate Synthesis, State Preparation, and Quantum Simulations

Swathi S. Hegde1, K. R. Koteswara Rao2 and T. S. Mahesh1∗1Department of Physics and NMR Research Center,

Indian Institute of Science Education and Research, Pune 411008, India2Fakultat Physik, Technische Universitat Dortmund, D-44221 Dortmund, Germany

A key task in quantum computation is the application of a sequence of gates implementing a spe-cific unitary operation. However, the decomposition of an arbitrary unitary operation into simplerquantum gates is a nontrivial problem. Here we propose a general and robust protocol to decomposeany target unitary into a sequence of Pauli rotations. The procedure involves identifying a com-muting subset of Pauli operators having a high trace overlap with the target unitary, followed bya numerical optimization of their corresponding rotation angles. The protocol is demonstrated bydecomposing several standard quantum operations. The applications of the protocol for quantumstate preparation and quantum simulations are also described. Finally, we describe an NMR exper-iment implementing a three-body quantum simulation, wherein the above decomposition techniqueis used for the efficient realization of propagators.

I. INTRODUCTION

Quantum devices have the capability to perform sev-eral tasks with efficiencies beyond the reach of their clas-sical counterparts [1, 2]. An important criterion for thephysical realization of such devices is to achieve pre-cise control over the quantum dynamics [3]. The circuitmodel of quantum computation is based on the realiza-tion of a desired unitary in terms of simpler quantumgates. However, arbitrarily precise decomposition of ageneral unitary UT in the form

UT = Um · · ·U2U1, (1)

is a nontrivial task. Here each of the Uj ’s is either oflower complexity or acts on smaller subsystems. Such adecomposition is said to be efficient if (i) m scales poly-nomially with the system size n and (ii) spatial and tem-poral overhead of each Uj scales polynomially with n.

The decomposition of a unitary operator correspond-ing to a HamiltonianH = HA+HB, where [HA,HB] = 0,is trivial, i.e., UT = e−iHt = e−iHAte−iHBt. When[HA,HB] 6= 0, one can discretize the time, δ = t/m,and use the Trotter’s formula [4]

UT =[e−iHAδe−iHBδ

]m+O(δ2) (2)

or its symmetrized form [5]

UT =[e−iHAδ/2e−iHBδe−iHAδ/2

]m+O(δ3). (3)

For a time-dependent HamiltonianH(t), one needs to useDyson’s time ordering operator or the Magnus expansion,and then decompose the time discretized components [6].However, for a given unitary UT , such a decomposition

∗Electronic address: mahesh.ts@iiserpune.ac.in

may not be obvious or even after the decomposition, theindividual pieces themselves may involve matrix expo-nentials of non-commuting operators thus failing to re-duce the complexity.

Several advanced decomposition routines have beensuggested for arbitrary unitary decomposition. Barencoet al. have shown that XOR gates along with local gatesare universal, and in terms of these elementary gatesthey have explicitly decomposed several standard quan-tum operations [7]. Tucci presented an algorithm to de-compose an arbitrary unitary into single and two-qubitgates using a mathematical technique called CS decom-position [8]. Khaneja et al. used Cartan decompositionof the semi-simple lie group SU(2n) for the unitary de-composition [9]. A method to realize any multiqubitgate using fully controlled single-qubit gates using Greycode was given by Vartiainen et al. [10]. Mottonen etal. have presented a cosine-sine matrix decompositionmethod synthesizing the gate sequence to implement ageneral n-qubit quantum gate [11]. Recently, Ajoy et al.also developed an ingenious algorithm to decompose anarbitrary unitary operator using algebraic methods [12].More recently, this method was utilized in the experimen-tal implementation of mirror inversion and other quan-tum state transfer protocols [13]. Manu et al have shownseveral unitary decompositions by case-by-case numericaloptimizations [14].

In this article, we propose a general algorithm to de-compose an arbitrary unitary upto a desired precision.It is distinct from the above approaches in several ways.Firstly, our method considers generalized rotations ofcommuting Pauli operators which are more amenablefor practical implementations via optimal control tech-niques. Secondly, being a numerical procedure, it con-siders various experimentally relevant parameters suchas robustness with respect to fluctuations in the controlparameters, minimum rotation angles etc. Besides, theprocedure can be extended for quantum circuits, quan-tum simulations, quantum state preparations, and prob-

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22

Mar

201

6

2

W0 = 1

Rj = UTWj-1†

fj = Tr[Rj Pj()]

{Pj , j}

is the PDCS

Fj > Fth ?

Select Pj having maximum fj and

construct Wj = Vj…V1

YES NO

j = Wj-1 0Wj-1†

fj = Tr[j Pj() T]

Fj = Tr[UTWj†] Fj = Tr[j½ T j½]

optimize j by maximizing Fj

NO

W0 = 1

Algorithm for a unitary decomposition

Algorithm for state preparation

FIG. 1: (Color online) The flowchart describing PDCS algo-rithm for unitary decomposition (left) and state preparation(right, dashed).

ably in some cases even for nonunitary synthesis.

The paper is organized as follows: A detailed explana-tion of the algorithm for arbitrary unitary decompositionis presented in section II . Section III deals with the ap-plications of the algorithm with explicit demonstrationsinvolving the decomposition of standard gate-synthesis,quantum circuit designs, certain quantum state prepara-tions, and for quantum simulations. Finally we concludein section IV.

II. ALGORITHM

In the following, we describe an algorithm for Paulidecomposition over commuting subsets (PDCS) for arbi-trary unitary operators. Although for the sake of sim-plicity we utilize 2-level quantum systems, the protocolapplies equally well to any d-level quantum systems.

Let UT be the desired unitary operator of dimensionN = 2n to be applied on an n-qubit system. We seekan m-rotor decomposition Wm = VmVm−1 · · ·V1 ≈ UT ,where the decomposed unitaries Vj with j ∈ [1,m] havethe form

Vj = e−i∑β P

(β)j φ

(β)j . (4)

Here {P (β)j } ≡ Pj is a maximally commuting subset of

n− qubit Pauli operators, {φ(β)j } ≡ Φj is the set of corre-sponding rotation angles, and the index β runs over theelements of Pj . In general, for an n-qubit case, a maximalcommuting subset Pj can have at most N − 1 elements.The fidelity Fm of the decomposition, defined by

Fm = 〈UT |Wm〉 = |Tr[U†TWm]/N |, (5)

should be larger than a desired threshold Fth.The flowchart for the PDCS algorithm is shown in Fig.

1. We now describe an algorithm to build Wm in m steps.To begin with, we start with W0 = 1. The jth step ofthe algorithm consists of the following processes:

1. Calculate the residual propagator Rj = UTW†j−1.

2. Selection of the commuting subset Pj having the

maximum overlap fj =∑β Tr[RjP

(β)j ] with the

residual unitary Rj .

3. Setting up the decomposition Wj and numeri-cally optimizing the rotation angles {Φ1, · · · ,Φj}by maximizing the fidelity Fj = 〈Wj |UT 〉, whereWj = Vj · · ·V1.

X /2 Y/4

1X/4

Z1/4 ZX/4

H (a)

(b)

ZZX 3/8

ZZ1 3/8

Z1X /8

11X /8 1Z1 3/8

1ZX 3/8

Z11 /8

(g)

ZZZ 3/8

ZYY /8

ZXX /8

1XX /8 1YY /8

1ZZ 3/8

Z11 3/8

X

X

(i)

(j) XXX 3/8

XX1 3/8

X1X 3/8

11X 3/8 1X1 3/8

1XX 3/8

X11 3/8

Grov

1Z/8

Z1/8 ZZ/8

(d)

S

1Z/4

Z1/4 ZZ /4

(c)

Z

Z

S

XX/4

YY/4 ZZ/4

X

X

(e)

Swap

Grov

1X/4

G X1/4 XX/4

(f)

ZZZ /8

ZZ1 /8

Z1Z /8

11Z /8 1Z1 /8

1ZZ /8

Z11 /8

(h)

Z

FIG. 2: (Color online) PDCS of some standard quantumgates: (a) single-qubit Hadamard, (b) c-NOT, (c) c-Z, (d) c-S,(e) SWAP, (f) 2-qubit Grover iterate, (g) Toffoli, (h) c2-Z, (i)Fredkin, and (j) 3-qubit Grover iterate. The individual rotorsare represented by dots, triangles, and heptagons dependingon number of Pauli operators (indicated at the vertices) ineach rotor. The corresponding rotation angles are indicatedby subscripts.

3

(c)

AQFT

(b)

H4 H3 H2 H1

QFT

(a)

S H2 H1

S34 S23 S12

11X /2

Z11/4

1XX/4

ZX1/2

ZXX/4

3,5,6

ZXZ 3/8

ZX1 /8

Z1Z 3/8

11Z 3/8

/8

1XZ 3/8

Z11 /8

2,4,6

ZZX /8

ZZ1 3/8

Z1X 3/8

11X /8

1Z1 /8

1ZX 3/8

Z11 /8

2,5,7 IQFT1,2,3 H1,2,3

1X1

FIG. 3: (Color online) PDCS of (a) 2-qubit Quantum Fourier Transform (QFT), (b) 4-qubit approximate QFT (AQFT), and(c) 7-qubit Shor’s circuit for factorizing 15. In (b), each S gate acts on a pair of qubits as indicated by the subscripts.

These steps can be iterated upto m-steps until the fidelityFm > Fth of a desired value is reached.

In general, the solutions to the decomposition may notbe unique. However, it is desirable to attain a decompo-sition that is most suitable for experimental implementa-tions. In this regard, we look for solutions with minimumrotation angles {Φj}, which can be obtained by using asuitable penalty function in step 3 of the above algorithm.

III. APPLICATIONS

A. PDCS of quantum gates and circuits

In this section we illustrate PDCS of several standardquantum gates. As described in Eq. 4, the jth decom-position is expressed in terms of the commuting Paulioperators Pj and the corresponding rotation angles Φj .

Further, a specific operator P(β)j ∈ Pj can be expressed

as a tensor product of single-qubit Pauli operators X, Y ,Z, and the identity matrix 1.

Exact PDCS of several standard quantum gates areshown in Fig. 2. For a single-qubit Hadamard opera-tion (Fig. 2(a)), we obtain a decomposition with twononcommuting rotations, as is well known [1]. HereP1 = X, P2 = Y and the corresponding rotation anglesΦ1 = −π/2, Φ2 = −π/4, are indicated by the subscripts.

In the two-qubit case, the maximal commuting subsetcan have only three Pauli operators and there are only15 such subsets. Figs. 2(b-e) describe decompositionsof several two-qubit gates namely c-NOT, c-Z, c-S, and

SWAP gate. Here Z =

[1 00 −1

]and S =

[1 00 i

]. It is

interesting to note that each of these gates needs a singlesubset of commuting Pauli operators. Such a rotationcan be obtained by a single matrix exponential and canbe thought of as a single generalized rotation in the Paulispace. We refer to such a generalized rotation as a rotor,and since it consists of three operators, we represent it

by a triangle. In practice, the individual components of asingle rotor can be implemented either simultaneously, orin any order. We find that even a 2-qubit Grover iterate,i.e., G = 2|ψ〉〈ψ| − 1, where |ψ〉 = (|00〉 + |01〉 + |10〉 +|11〉)/2, can be realized as a single rotor (Fig. 2(f)).

For three-qubits, the maximal commuting subset canhave only seven commuting Pauli operators and thereare 135 distinct subsets. Figs. 2(g-j) describe Toffoli,c2-Z, Fredkin, and 3-qubit Grover iteration respectively.Again we find that a single heptagon rotor suffices forrealizing each of the standard gates. Similarly, in the caseof four-qubits, a maximal commuting subset can have 15operators and one can verify that a basic gate such asc3-NOT, c3-Z, etc. can be realized by a single rotor.

It is always possible to decompose a multi-qubit quan-tum circuit in terms of single- and two-qubit gates [7].As examples, we shall consider PDCS of a few quantumcircuits (Fig. 3). The two-qubit Quantum Fourier Trans-form (QFT) circuit can be exactly decomposed into threerotors as shown in Fig. 3(a). As another example, PDCSof the 4-qubit approximate QFT (AQFT) circuit [15] re-sults in only single-qubit and two-qubit rotors as shown inFig. 3(b). Similarly, PDCS of the 7-qubit Shor’s circuitfor factoring the number 15 involves at most three-qubitrotors as shown in Fig. 3(c) [16]. In this sense, PDCS ofmultiqubit quantum gates and quantum circuits is scal-able with increasing system size.

B. Quantum state preparation

Here the goal is to prepare a target state ρT start-ing from a given initial state ρ0. In general the uni-tary operator, connecting the initial and target states,itself is not unique. The procedure is similar to thatdescribed in the previous section, and is summarized inthe flowchart shown in Fig. 1. Here the selection ofcommuting Pauli operators Pj is based on the overlap

fj =∑β Tr[ρjP

(β)j ρT ], where ρj = Wjρ0W

†j is the in-

termediate state after jth decomposition. As explained

4

YX/8

XY/8

(a)

(b)

(d)

(c)

00 00 + 11

2

ZY/4

YZ/4 X1 1X

XXY/4

000 000 + 111

2

000

1XX/3

X1X/3 XX1/6

XXX/2 XZX/3

YY1/6 ZXX/3

001+010 +100

3

FIG. 4: (Color online) PDCS of some state to state trans-fers: (a) polarization transfer (INEPT) and (b-d) preparationof Bell, GHZ, and W states respectively starting from purestates.

before, we select the commuting subset Pj having themaximum overlap fj and optimize the phases Φj by max-imizing the Uhlmann fidelity

Fj = 〈ρj |ρT 〉 = Tr[ρ1/2j ρT ρ

1/2j ]. (6)

Again, m iterations are carried out until Fm ≥ Fth isrealized.

Fig. 4 displays PDCS of some standard state to statetransfers. The polarization transfer in a pair of qubits(popularly known as INEPT [17]) requires a single ro-tor having a pair of bilinear operators (Fig. 4(a)). Thepreparation of a Bell and GHZ states respectively from|00〉 and |000〉 states also require a single rotor (Fig. 4(b-c)). However, the preparation of a three-qubit W-state issomewhat more elaborated, and requires two rotors (Fig.4(d)). Although these decompositions are not unique, itis possible to optimize them based on the experimentalconditions. Here one can notice that although a maximalcommuting subset can have up to N − 1 elements, it isoften possible to decompose a multi-qubit operation overa smaller commuting subset.

J

JHC = 224.5

JFC = 310.9

JHF = 49.7

1H

19F 13C

T1 T2*

H 13.7 0.4

F 5.2 0.2

C 1.9 0.5

FIG. 5: (Color online) Dibromofluoromethane consisting ofthree nuclear spin qubits 1H, 13C and 19F. The tables displaythe values of indirect spin-spin coupling constants (J) in Hz,and the relaxation time constants (T1, and T ∗

2 ) in seconds.

C. Quantum simulation

Utilizing controllable quantum systems to mimic thedynamics of other quantum systems is the essence ofquantum simulation [18]. Various quantum devices havealready demonstrated quantum simulations of a numberof quantum mechanical phenomena (for example, [19–22]). An important application of the decompositiontechnique described above is in the experimental realiza-tion of quantum simulations. To illustrate this fact, weexperimentally carryout quantum simulation of a three-body interaction Hamiltonian using a three-qubit system.While such a Hamiltonian is physically unnatural, simu-lating such interactions has interesting applications suchas in quantum state transfer [23]. Specifically, we simu-late the dynamics under the Hamiltonian

HS = X11 + 1X1 + 11X + J123ZZZ, (7)

where J123 is the three-body interaction strength. Aslightly different three-body Hamiltonian simulated ear-lier by Cory and coworkers [24] consisted of only the lastterm. The presence of other terms which are noncom-muting with the 3-body term necessitates an efficient de-composition of the overall unitary.

We use three spin-1/2 nuclei of dibromofluoromethane(Fig. 5) dissolved in acetone-D6 as our three-qubit sys-tem. All the experiments are carried out on a 500 MHzBruker NMR spectrometer at an ambient temperatureof 300 K. In the triply rotating frame at resonant offsets,the internal Hamiltonian of the system is given by

Hint = [JHFZZ1 + JHCZ1Z + JFC1ZZ]π/2, (8)

and the values of the indirect coupling constants{JHF, JHC, JFC} are as in Fig. 5. This internal Hamilto-nian along with the external control Hamiltonians pro-vided by the RF pulses are used in the following to mimicthe three-body Hamiltonian in Eqn. 7.

The traceless part of the thermal equilibrium state ofthe 3-qubit NMR spin system is given by ρeq = (Z11 +1Z1 + 11Z)/2 [17]. The initial state ρ(0) = (X11 +1X1 + 11X)/2 is prepared by applying a 90◦ RF-pulseabout Y . The goal is to subject the three-spin systemto an effective three-body Hamiltonian HS and monitorthe evolution of its state ρ(t) = US(t)ρ(0)US(t)†, whereUS(t) = e−iHSt. We choose to experimentally observe

ZZ1

Z1Z /2

YYX 3.0

0.48

XYY 3.0

YXY 1.72

XXY 3.0

XYX 0.16 YXX 3.0

XXX

FIG. 6: (Color online) PDCS of Us(τ) = exp(−iHsτ) forJ123 = 5 Hz and τ = 0.8 s.

5

0 2 4 6 8 10 12 14 16

0

0.5

t (s)

Mx (

t)

1 2 3 4 50.2

0.4

0.6

0.8

1

m

F

Trotter (Symmetrized)

Trotter

Numerical Pauli decomposition

Exact PDCS Experiments

FIG. 7: (Color online) Magnetization vs time.

the transverse magnetization

Mx(kτ) = Tr[ρ(kτ)(X11 + 1X1 + 11X)/2] (9)

for J123 = 5 Hz at discrete time intervals kτ , where k ={0, · · · , 20} and τ = 0.8 s.

The PDCS of US shown in Fig. 6 consists of two rotors:a hexagon followed by a triangle. We utilized bang-bang(BB) control technique for generating each of the two ro-tors [25]. The duration of each BB-sequence was about 7ms and fidelities were above 0.98 averaged over a 10% in-homogeneous distribution of RF amplitudes. The resultsof the experiments (hollow circles) and their comparisonwith numerical simulation of the PDCS (triangles) andexact numerical values (stars) are shown in Fig. 7. Thefirst experimental data point was obtained after a sim-ple 90 degree RF pulse and was normalized to 1. Thekth point is obtained by k iterations of the BB-sequencefor US(τ). While the experimental curve displays thesame period and phase as that of the simulated curve,the steady decay in amplitude is mainly due to decoher-ence and other experimental imperfections such as RFinhomogeneity and nonlinearities of the RF channel.

To compare the efficiency of PDCS with that of Trotterdecomposition (in Eq. 2 and 3), we calculate the fidelities(F ) of the decomposed propagator with the exact propa-gator US(τ) as a function of number m of rotors (see Fig.8). It can be observed that, with increasing number ofrotors, PDCS fidelity converges faster than the Trotter.

Number of rotors0 5 10 15

Fid

elity

0.2

0.4

0.6

0.8

1

Trotter (Symmetrized)TrotterPDCS

FIG. 8: (Color online) Fidelity (Fm) vs number (m) of rotorsin the decompositions.

IV. CONCLUSIONS

In this work we proposed a generalized numerical al-gorithm based on Pauli decomposition over commutingsubsets (PDCS). The aim of the algorithm is to decom-pose an arbitrary target unitary into simpler unitaries,referred to as rotors. Each rotor consists of only commut-ing subset of Pauli operators. These rotors are optimizedto be robust against experimental errors by minimizingthe rotation angles and by considering other control er-rors. Thus apart from providing an intuitive and topolog-ical representation of an arbitrary quantum circuit, themethod is also useful for its efficient physical realization.

We demonstrated the robustness and efficiency of thedecomposition using numerous examples of quantumgates and circuits. It is interesting to note that sev-eral standard quantum gates correspond to single rotors.We also discussed the applications of PDCS in quantumstate-to-state transfers and illustrated it using several ex-amples.

Another important application of PDCS is in quan-tum simulations. As an example, we described the quan-tum simulation of a three-body interaction. We usedPDCS algorithm to decompose the unitary correspond-ing to such a Hamiltonian and found it to be more effi-cient than Trotter decomposition in terms of the numberof rotors. Further, we have demonstrated the quantumsimulation by experimentally monitoring the evolutionof magnetization using a three qubit NMR system. Theexperimental results matched with the numerical simu-lations upto a decay factor arising predominantly due todecoherence.

We believe such unitary decomposition strategiescombined with sophisticated optimal control techniqueswill greatly assist in efficient quantum control.

Acknowledgments

We thank Prof. Anil Kumar, IISc, Bangalore forproviding us the three-qubit NMR register. We alsoacknowledge useful discussions with Sudheer Kumar.This work was partly supported by DST Project No.DST/SJF/PSA-03/2012-13.

6

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