Adiabatic quantum computing with spin qubits hosted by molecules.

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Adiabatic Quantum Computing with Spin Qubits Hosted by Molecules Satoru Yamamoto, Shigeaki Nakazawa, Kenji Sugisaki, Kazunobu Sato,* Kazuo Toyota, Daisuke Shiomi, and Takeji Takui * 5

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Received (in XXX, XXX) Xth XXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XX DOI: 10.1039/b000000x A molecular spin quantum computer (MSQC) requires electron spin qubits which pulse-based electron spin/magnetic resonance (ESR/MR) techniques can afford to manipulate for implementing quantum gate operations in open shell molecular entities. Importantly, nuclear spins which are topologically connected, particularly in organic molecular spin systems are client qubits, while electron spins play a role of bus qubits. Here, we introduce the implementation for an adiabatic quantum algorithm, suggesting the possible utilization of molecular spins with optimized spin structures for MSQCs. We exemplify the utilization of an adiabatic factorization problem of 21, comparing with the corresponding nuclear magnetic resonance (NMR) case. Two molecular spins are selected: One is a molecular spin composed of three exchange-coupled electrons as electron-only qubits and the other an electron-bus qubit with two client nuclear spin qubits. Their electronic spin structures are well characterized in terms of the quantum mechanical behaviour in the spin Hamiltonian. The implementation of adiabatic quantum computing/computation (AQC) has, for the first time, been achieved by establishing electron spin resonance/magnetic resonance (ESR/MR) pulse sequences for effective spin Hamiltonians in a fully controlled manner of spin manipulation. The conquered pulse sequences have been compared with the NMR experiments and shown that much faster CPU times corresponding to the interaction strength between the spins. Significant differences are shown in rotational operations and pulse intervals for ESR/MR operations. As a result, we suggest the advantages and possible utilization of the time-evolution based AQC approach for molecular spin quantum computers and molecular spin quantum simulators underlain by sophisticated ESR/MR pulsed spin technology.

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Introduction

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In the current era, quantum behavior of synthetic molecular systems, especially focussed on synthetic open shell entities underlain by molecular optimization for quantum functioning, has attracted considerable attention from both the theoretical and experimental sides in chemistry, physics and materials science [1, 2]. This is due to the fact that spins explicitly arise from quantum nature. The interest in molecular quantum spin technology has developed with affording to manipulate and control more than a few molecular electron spins with sophisticated sequences of radiation pulses created by arbitrary wave-form generators (AWGs). Engineering or building-up such a large number of the pulses has been a formidable task until recently, if they are composed of more than three frequency-different microwave pulses or their relative phase control is required. By virtue of the AWG-based technical development, molecular spin quantum technology is emerging in chemistry, materials science and related fields of information science. This molecular quantum technology applied to molecular spins is pulse MW-mediated, implying that nuclear spin spins as client qubits (quantum bits) in molecular spins, while electron spins play the role of bus qubits, can be controlled not only by pulsed radiofrequency but also by pulsed MW techniques. The MW-mediated control of client qubits can afford to execute much faster quantum computation than only RF-mediated ones. Nowadays, classical computers (CCs) are essential sources

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for any analyses in pure and applied sciences. However, the total computational ability is intrinsically restricted by its physical reality, an ensemble of atoms involved [3], and computational approaches [4]. A quantum computation (QC) is a new paradigm in that bits relevant to quantum states are capable of processing quantum problems much faster since they utilize the superpositions of the states and/or their entanglement to speed up the processing [5,6]. Here, we only emphasize that QC, as expected, has expanded the computational ability from CCs, implying the ability of solving BQP (Bounded-error Quantum Polynomial time) classes in polynomial time, which contain some part of NP (Non-deterministic Polynomial time) classes [7]. Interesting problems are the intersection of the NP (or NP-hard) and BQP classes [8], and its most famous problem is Shor’s factorization algorithm [9], which has underlain the implementation of scalable and realistic QCs. Referred to truly realistic QCs, it is worth noting that performing quantum chemistry or quantum chemical calculations on QCs is one of the most important contemporary issues in applications of quantum information processing technology [10], in spite of the fact that to find any disruptive quantum algorithms for many-body fermion systems is still a challenging issue [11]. From an experimental viewpoint, the scalability of addressable qubits is the focus in terms of materials challenges [12]. Besides the implementation of practical universal QCs, synthetic molecular quantum simulators are an emerging issue, which can afford to solve intractable quantum problems of CCs [13].

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Figure 1. Schematic view of AQC. The blue and red lines indicate the ground and first excited states of a quantum system, respectively. The system Hamiltonian moves from the initial Hamiltonian (Hi) to the final one (Hf). g denotes a variable changing from 0 to 1 and corresponding to the initial state (g = 0) and final one (g = 1). The minimum energy gap is ∆E at g = gc.

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Since Shor’s algorithm appeared, many experimental attempts have been performed [14] and the first experiment was carried out by highly sophisticated pulsed NMR techniques [15]. The first experiment was made by manipulating C, H and F nuclear qubits of dimethylfluoromalonate molecule in solution. It has been claimed that quantum entanglement as the heart of QC was not been established in any NMR experiments in solution. Nevertheless, this very attempt brought QCs down to earth. Another factorization experiment was proposed by Peng et al., which factorizes 21 by an Adiabatic Quantum Computer (AQC) with only three qubits. AQC is a computation model in which quantum information is processed by utilizing the ground state of a quantum system with varying its Hamiltonian [16]. AQC is defined as the one different from a gate model quantum computer, which is composed of operations of single spin rotations and CNOT gates in standard cases. Between the two QCs, two important features are established: (1) AQC has the same computational ability as a standard QC [17] and (2) AQC can execute error corrections [18], which is of essential importance in quantum computation processes. The computational time executing Adiabatic Quantum Algorithm (AQUA) is defined by the energy gap ( ∆E ) between the ground state and the first exited state of the quantum systems under study. When ∆E and the problem size (N) have polynomial relationships, AQC can solve problems in polynomial time against the problem size. This is an important point in molecular optimization of realistic spin qubits in terms of chemistry. In general, a quantum algorithm and physical operations to execute the manipulation of qubits are related by particular transformation schemes. Many physically realizable qubits have been proposed such as molecular spin systems [2], photon qubits [19], trapped ions [20], quantum dots [21] and superconducting flux circuits [22]. Since AQUA is related to the time-dependent Hamiltonian path which is equivalent to the time evolution operator, the transformation is needed for physicochemical experiments. From the viewpoint of the ESR technique mentioned above, it is interesting to perform AQC on molecular spin qubits by using the MW-based ESR spin technology, making clear the difference from the NMR-QC approach. A molecular spin QC is defined as a system allowing us to manipulate spin qubits of an organic radical or open-shell entity by pulsed ESR techniques [23]. For quantum operations, the molecular spin QC This journal is © The Royal Society of Chemistry [year]

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utilizes hyperfine interactions whose strengths are three or four orders of magnitude large compared with those in the NMR-QC approach [24]. Molecular spin QC approaches require molecular optimization which renders electron or nuclear spins of molecular spins functioning as addressable spin qubits. The optimization includes g- or A-tensor engineering to distinguish between the qubits [12, 25, 26]. It is worth noting that the implementation of CNOT (Controlled-NOT) gates as two-qubit quantum gates has been for the first time achieved for synthetic molecular electron spins [26], and chemistry emerges in the field of quantum computing and quantum information processing (QC/QIP). Here, we present a theoretical study of AQC on molecular spin QCs equipped with only few spin qubits, treating with a factorization. A series of pulse sequences is implemented on the basis of real molecular spins and computational times are estimated.

Theoretical conditions for pulse sequences The factorization algorithm of AQC

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The algorithm of AQC equals the quantum path traced by the ground state of a system defined by a time-dependent Hamiltonian (Figure 1). This ideal Hamiltonian governing the quantum path of AQUA is accounted for/approached by the real Hamiltonian of qubits in molecules. Hereafter we focus on the former ideal one. Here, we discuss the former one. In the adiabatic factorization algorithm of 21, the final Hamiltonian (or defines as the problem Hamiltonian : Hˆ f ), whose ground state gives an answer of the factorization problem, is selected as Eq. (1-1) [15],

(

Hˆ f / h = Nˆ − xˆyˆ 75

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)

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Eq. (1-1)

where, Nˆ = 21Iˆ and (x, y) is the binary representation by the spin qubits, and the algorithm requires only three qubits such as xˆ = ( Iˆ − σ 1z ) + Iˆ , yˆ = 2( Iˆ − σ z2 ) + ( Iˆ − σ z3 ) + Iˆ . The classical version of this algorithm corresponds to the minimization problem of 2 f(x, y) = (N − xy ) . Obviously, the ground state of Hˆ f , in the classical case the minimized value for f(x, y) , gives the answer of the factorization problem of N, because the ground state (|↓↓↓>) allows one to calculate the values for (x, y). This final Hamiltonian is not efficient for a certain case when the solutions of x and y need the same bit length. In this case, there are two solutions of (x, y) and (y, x), then ∆E min = 0 . However, there is some possibility to calculate other problems fast, and at least this algorithm allows us to carry out AQC experiments by using a small number of qubits. Expanding Eq. (1-1), we obtain the following expression:

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Hˆ

f

h ~ 84 σ 1z + 88 σ

2 z

+ 44 σ

3 z

− 20 σ 1z σ

2 z

−10σ 1z σ z3 − 20σ z2 σ z3 − 16σ 1z σ z2 σ z3

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Eq. (1-2)

which is utilized for implementing pulse sequences. In this transformation, an identity operator Iˆ is neglected. To perform AQC, we need to define an adiabatic path including the starting point of the algorithm, an initial Journal Name, [year], [vol], 00–00

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Hamiltonian ( Hˆ I ). For comparison with the NMR-QC case, we define the following expressions in the same manner as in the NMR experiment, (Eqs. (1-3) to (1-4)) [15]. 40 3

Hˆ I / h = 30 ∑ σ

5

Eq. (1-3)

i x

i =1


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Eq. (1-4a)

Uˆ = ∑ exp( −0.028 × iHˆ m / h ) i =1

Hˆ m = ( m / 5) 2 Hˆ f + {1 − (m / 5) 2 }Hˆ m

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Eq. (1-4b)

In Eqs. (1-4), the time evolution operator of this algorithm was approximated in finite time steps for pulsed ESR experiments. In order to avoid the crossing path between the ground state and excited states, technically AQC needs a non-commutative time evolution operator Uˆ , therefore Uˆ should be transformed by the Trotter expansion into the commutative operators for the ESR experiments (Eqs. (1-5)). Uˆ =

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∏

Uˆ mi × Uˆ mf × Uˆ mi

Eq. (1-5a)

m =1

Uˆ mi = exp( −i 0.028{1 − (m / 5) 2 }Ηˆ i /2h)

Eq. (1-5b)

Uˆ mf = exp( −i0.028(m / 5) 2 Hˆ f /h)

Eq. (1-5c)

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In this approach, the theoretical fidelity is known to be 0.91 [15], and we adopted this time evolution operator as the simulated operator in the implementation of pulse sequences under study.

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between spins are described also by 2nd rank tensors of J, D and A, which denote exchange, spin-dipolar and hyperfine interactions, respectively. The spin Hamiltonian for single crystal systems is approximated to the effective spin Hamiltonian by the following procedure. The effective spin Hamiltonian with the static magnetic field along the z-direction was calculated from Eq. (2-1) by adopting an individual rotational frame and a secular averaging approximation with small anisotropy of g-tensors [27]. The molecules hosting spin qubits are assumed to be embedded in lattices of single crystals, since most of QC experiments on molecular spin QCs have been carried out in solid states and such ensemble experiments enable us to establish both qubit initialization and entanglement conditions [23,28]. For simplicity, the static magnetic field is oriented parallel to the principal axis of all the hyperfine tensors. Then the effective spin Hamiltonian in the time evolution operator can be described as Eqs. (3-1) and (3-2).

The Hamiltonian of molecular spin QCs

3e 2 2 23 3 3 31 1 Hˆ int = S 1z ( j + D )12 zz S z + S z ( j + D ) zz S z + S z ( j + D ) zz S z

Eq. (3-1)

1e + 2n 12 2 3 3 31 1 Hˆ int = S 1z Azz I z + I z2 ( j + D) 23 zz I z + I z Azz S z

Eq. (3-2)

where ( j + D)ijzz and Azzij are the z,z-components of the tensors. In solution-state ESR cases, D tensors vanish due to their traceless character and the restriction between the static magnetic field and hyperfine tensors also disappears. The effective spin Hamiltonian Eqs (3-1) and (3-2) are adopted for implementing pulse sequences, which simulate the time evolution operators of AQUA, Eqs (1-2), (1-3) and (1-5). Details on this procedure are given in Supporting Information.

The spin Hamiltonian of real molecule spin qubit system is written as Eq. (2-1) in Schrödinger picture. The molecular system and spin interaction parameters Hˆ MSQC =

N

N ,N

M

∑ S g β B − ∑ I g β B + ∑ S ( j + D) i

i =1

i e

i e

i

i n

i n

i

i =1

N,M 30

+

Sj 70

M ,M

∑ S A I + ∑ I ( j + D) i

i = j =1

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ij e

i< j

ij

i

i

ij n

Ij

Eq. (2-1)

i< j

where, N and M denote the number of electrons and nuclei, respectively. The first and second terms are Zeeman interactions and g i is a 2nd rank tensor which is related to the Larmor frequency ω0 of an ith spin ( ω0i = g zzi β i Bz / h where Bz denotes the static magnetic field in the system). The interactions

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A phthalocyanine derivative [29] and a 2-deuterated glutaconic acid radical [30,31] are adopted for molecular spin QCs as three electron qubit (3e) system and one electron and three nuclear client qubit (1e+2n) system, respectively (Figure 2). These systems are selected for the experimental reasons. A phthalocyanine system with four electron spins has been reported [29]. In this study, we assumed that one of the four nitroxide radical ( N − O ⋅ ) sites is reduced to form the closed shell of N − O − H . The unpaired electrons (e1, e2 and e3) are numbered in Figure 2. The glutaconic acid radical is known to have

Figure 2. Molecular systems for AQC. a) A phthalocyanine system for three electron spin qubits. One radical site is reduced by a hydrogen atom. Three electron spins are mostly localized on each NO radical site, which is numbered in red. b) A trans-glutaconic acid radical molecule for one electron spin bus qubit and two nuclear client qubits. One hydrogen atom is deuterated and the number 1 denotes an unpaired electron spin and the numbers 2 and 3denote the client qubits. It should be noted that the unpaired electron is delocalized to some extent over the π-conjugation


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approximately the collinearity among the hyperfine tensors [30], which is essential for the present pulse sequence study, and we assumed its deuterated derivative. The spin qubits (e1, H2 and H3) is depicted in Figure 2. The spin Hamiltonian parameters of Eqs. (3-1) and (3-2) were obtained by quantum chemical calculations for the phthalocyanine system and those for the glutaconic acid system from the X-ray irradiation study [30]. In the quantum chemical calculations, the molecular structure was optimized and was of Cs symmetry by the UB3LYP/6-31G* and the J coupling parameters were calculated by means of the broken symmetry density functional theory (BS-DFT) method in Gaussian03 [32,33]. Note that the BS scheme is originally proposed for two-spin based singlet-triplet systems and the extension of the BS-DFT approach to three-spin based triradical systems is not trivial. Details of the calculations for the present three-spin system are given in Supporting Information, emphasizing the non-trivialities of the straightforward extension. Since the phthalocyanine derivative has localized electrons with the spin distance larger than 10 Å, the D tensors were calculated by the point dipole approximation with the optimized structure.

Table 1. Coupling constants obtained by quantum chemical calculations for the phthalocyanine system. a) Coupling constants of the J tensors b) Spin distances and coupling constants of the D tensors. a) e2- e3 e1- e3 e1- e2 J/h /MHz -12.01 -12.01 -66.03 b) r /Å D/h /MHz Dzz/h /MHz

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e1- e2 13.31 -33.11 -16.55

e2- e3 13.31 -33.11 -16.55

The single qubit manipulations appearing in Uˆ mi and the single qubit operation terms ( σ z ) of Uˆ mf are relevant to the only spin rotations of the x- and y-axis directions. The Uˆ mi term is replaced with the single spin rotations around the x-axis direction since Uˆ mi is composed of only the sum of the σ x operations (Eq. (4-1)). For the same reason, the single qubit operations term ( Uˆ ms f ) in Uˆ mf can be replaced with the certain angle rotation of the y-axis direction and the x-axis direction for changing the operators σ y to σ z (Eqs. (4-2)). 3

∑σ

Uˆ mi = exp(−i 0.42{1 − (m / 5) 2 }

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Parameter sets of the three electron spin qubit system The calculated values for the tensors are given in Table 1, which are adopted for implementing pulse sequences. The magnetic field direction is oriented along the z-axes depicted in Figure 2 because the orientation gives suitable interaction strengths for the phthalocyanine derivative and the anisotropic terms of the hyperfine tensors for the deuterated glutaconic acid system, respectively. The adopted hyperfine coupling constants of the 31 glutaconic acid system are A12 zz = +7 .0 MHz and Azz = −37.9 MHz [30]. In this electron spin bus qubit, the exchange interaction between the nuclei was estimated to be much smaller than the hyperfine interactions, therefore any operation relevant to this interaction is not included in the present pulse sequences.

i x)

Eq. (4-1)

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Results and Discussion 25

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Uˆ mf = exp(−i0.028(m / 5) 2 (84σ 1z + 88σ z2 + 44σ z3 ))

Eq. (4-2a)

 π   π  = exp − i σ xk  ⋅ exp(−iασ ky ) ⋅ exp i σ xk   4   4 

Eq. (4-2b)

exp(−iασ zk )

On the other hand, the highest-order spin operation, the 3-qubit interaction in Uˆ mf , can be performed by 2-qubit operations and single qubit rotations [34] (see Supporting Information for details). This method is applicable to n-spin ( n ≥ 3 ) operations, therefore the remaining problem is to create the 2-qubit operations between two qubits. In the three electron spin qubit system, there is a relationship associated with the effective spin Hamiltonian ( Hˆ eff = ∑3i < j α ijσ ziσ zj ). Let us assume the conditions, i ≠ k , k ≠ j, j ≠ i, −k exp(−iα ijσ zi σ zi t / h) = exp(−iHˆ eff t / 2h) ⋅ exp(−iHˆ eff t / 2h)

 π   π  −k exp(−iHˆ eff t / 2h) = exp − i σ xk  ⋅ exp(−iHˆ eff t / 2h) ⋅ exp i σ xk   2   2 

Eq. (4-4) Eq. (4-5)

The generation procedure for implementing pulse sequences 40

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The pulse sequence for the ESR-QC experiments was analytically calculated on the basis of individual spin manipulations. The operation set based on the ESR techniques is assumed as arbitrary spin rotations (x- and y-axis directions) for each spin and the time 3e 1e + 2n evolution with the effective spin Hamiltonian ( Hˆ int or Hˆ int ). In this calculation, we replaced Uˆ mi and Uˆ mf of Eq. (1-5a) with the following pulse operations. Here, Uˆ mi contains only single qubit operations of σ x . On the other hand, Uˆ mf contains single qubit operations, 2-qubit operations and 3-qubit operation of σ z which are associated with Hˆ f (Eqs. (1-2) and (1-3)). Since every term of Hˆ i (or Hˆ f ) is commutable with the other terms, the time evolution of Uˆ mi ( Uˆ mf ) is separable and corresponds to every term of Hˆ i ( Hˆ f ).


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−k = α ijσ zi σ zj |i ≠ k ≠ j −∑i ≠ k α ik σ zi σ zk , and Eq. (4-5) can where Hˆ eff 3

also be represented by the spin rotations of the y-axis direction. Another problem can be seen in 1e+2n systems such as the glutacoic acid radical for the spin interaction between the nuclei. This interaction is three or four orders of magnitude smaller than those from hyperfine interactions, then it is needed to replace it with a fast operation which can be composed of the 3-qubit operation and the other 2-qubits ones. All pulse operations are gathered to build up the total pulse sequences. In this study, we connected the pulses only when there are neighbouring operations relevant to the same spins and directions. The detailed analytical calculations are given in Supporting Information.


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Figure 3. Conquered pulse sequences for AQC. Time intervals are given in the upper of the panel of each pulse sequence. Numbered pulse operations in blue denote rotations around the x- or y-axis direction. a) A pulse sequence in the phthalocyanine derivative for the factorization of 21. The sequence indicates one cycle for m (ranging from 1 to 5), therefore this sequence is needed to loop at five times along the purple arrows. b) A pulse sequence in the glutaconic acid system for the factorization of 21. The sequence indicates one cycle for m (ranging from 1 to 5), therefore this sequence is needed to loop at five times along the purple arrows. c) Schematic picture for the operations appearing in the pulse sequences. Narrow and wide pulses denote the π/2 and π angle rotations, respectively. Black and red blocks denote the operations for the x- and y-axis direction, respectively.

The operation time of pulse sequences

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Conquered pulse sequences of AQC on the phthalocyanine (3e) derivative and the glutaconic acid radical (1e+2n) system are shown in Figure 3. The detailed parameters for the pulse intervals and the rotational angles required for solving the factorization of 21 are given in Table 2. To claim any practical scheme of quantum computation, the spin qubit manipulation based on the pulse sequence must be executed in polynomial time. The sequence generation method for AQC, which is also useful for quantum simulation by using molecular spins, gives the simplest procedure and fulfills the requirement for the computation time. The required time was estimated as the total time of the pulse intervals, as shown in Table 3. The phthalocyanine and glutaconic acid systems need 0.176 µs and 1.31 µs to complete the factorization, respectively. In the previous NMR-QC experiment on the closed shell molecule, the corresponding required time was approximately 50 ms. The time length strongly depends on the strength of interactions used for pulse operations and exchange coupling constants between the nuclei ranging from This journal is © The Royal Society of Chemistry [year]

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50 to 200 Hz. In comparison with the NMR-QC case with three nuclear spin qubits, both the ESR-QC molecular spins, the 3e and 1e+2n systems give about 7 up to 55 MHz for the interaction strengths, and thus the 3e and 1e+2n systems can speed up performing AQC-base factorization about 2.8 × 105 and 3.8 × 10 4 times faster than the NMR-QC case, respectively. Besides initialization issues of spin qubits in ensemble, in this context molecular electron spins have a big advantage over NMR-QC cases. As expected for ESR-QC experiments based on the circuit model [26], the present AQC approach demonstrates that hyperfine interactions with reasonable strength insure marked speedup in performing QC processes.

The operation angles of pulse sequences 35

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Referred to the spin manipulation appearing in the pulse sequences, a larger total number of pulses have to be executed, 140 pulses for the 3e system and 240 pulses for the 1e+2n system, than the NMR-QC case with 95 pulses [15], and accordingly the operation angles increased (Table 3). The operation angles relevant to the three spin qubits are 29.8π, 34.9π, 28.2π in units of Journal Name, [year], [vol], 00–00

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radian for the 3e system, and 39.8π, 70.0π, 53.2π for the 1e+2n system, while 21.5π, 24.0π, 11.5π for the NMR-QC case with three nuclear spin qubits. The required time is fast enough for the ESR-QC pulse sequences, and thus the total operation angles are not a serious issue. The rotation angles depend on the methods for generating the pulse sequences. A straightforward method for the sequence generation in the 3e system takes approximately 1.5 times larger time of the pulse operation/spin rotation angles than the NMR-QC case. Compared to the 3e system, the operation angles in the 1e+2n system result from approximately twice a larger number of the pulse operations than the 3e-system. This is due to the replacement techniques of the spin interaction between the nuclei. This indicates that the spin rotation angles of the present AQUA are governed by the sequences composed of 3-qubit operations. Physically, the 3-qubit operation needs three 2-qubit operations in addition to single qubit operations. Another important aspect is found for the 2-qubit operations in the 1e+2n system. In the present approach, at least the 3-qubit operation is required for creating the 2-qubit operation between the nuclei. This process decreases the required time as the same order of magnitude as the 3e system, and thus increases the number of the pulse operations and the amount of the rotation angles. This is a negative aspect intrinsic to the present approach and influential in the case that a final Hamiltonian is processed by only a small number of spin qubit operations.

Table 3. Required times and total operation angles in units of radian. a) Required time and total operation angles for the phthalocyanine derivative. b) Those for the glutaconic acid radical. a) e1 e2 e1 Operation angles 29.8π 35.0π 28.2π Required time 0.176 µs b)

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n3 53.2π

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The issue above appears particularly in AQC, but not in gate model QC approaches. Simulating the adiabatic path demands larger numbers of operators than the standard QC, whose processes are composed of only single spin rotations and CNOT-gate (two-qubit gate) operations. We note that this is an essential issue for performing AQC in the present scheme with molecular spin qubit systems. As a result, the pulse rotations of AQC should be treated elaborately for ESR-QC systems. In molecular electron spins in ensemble, relevant Rabi operations can be performed in the order of sub-microsecond [26]. The operation time is in the same order as a typical short period of the spin operation, then the generation of the pulse forms can be a significantly tough task in current pulse MW technology. On the other hand, the Rabi operation in NMR-QC cases (e.g. π pulses) takes much time (about a few µs for nuclei) than molecular electron spins [28], then one needs to eliminate the spin interactions while operating on nuclei. Based on the estimation from the numbers of the pulses and the Rabi frequencies, the required time for executing the AQC algorithm

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n2 70.0π 1.31 µs

paradigm is to implement correct and short time operations (below 0.1 ns) suitable for the ESR-QC system. In this work, we assumed that the ESR-QC experiments were carried out under the conditions of conventional MW frequencies at X-band. In principle, it is possible to set up ESR-QC experiments with the strong static magnetic field, which erase the anisotropic terms of the spin Hamiltonian in the secular averaging approach.

Discussion for ESR-QC experiments From the experimental viewpoint, we found important difference in the pulse intervals between the spin qubits under study. The pulse intervals between two spins are found to be much smaller in the ESR-QC systems (shorter than 1 ns: about 0.2 ns).This is because the interval time of the present algorithm is optimized for the interaction strength of NMR cases (Table 2b). Under this condition, we can execute operations of the pulse sequence with somewhat reduced fidelity or perform AQC with scaling up the pulse intervals (resulting in the longer required time). Another

e1 39.8π

Operation angles Required time

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Table 2. Detail parameters of the pulse sequences. a) Analytical time intervals of the pulse intervals for the phthalocyanine system and the glutaconic acid radical. b) Numerical time intervals (m = 1) of the pulse intervals for the phthalocyanine system and the glutaconic acid radical. c) Analytical operation angles and direction of the blue blocks (see Fig.3) in the pulse sequences. Here, bm = 0.028{1 − (m / 5) 2} and cm = 0.028(m / 5) 2 . a) t1 /ns t2 /ns t3 /ns t4 /ns t5 /ns phthalocyanine

− πh /( Dzz + J )13

− 64am /( Dzz + J )12

− 80am /( Dzz + J )12

− 40am /( Dzz + J )13

− 80am /( Dzz + J ) 23

glutaconic acid

− πh / A13 zz

64am / A12 zz

80am / A12 zz

− 40am / A13 zz

πh / A12 zz

b) phthalocyanine glutaconic acid

t1 /ns

t2 /ns

t3 /ns

t4 /ns

t5 /ns

9.158 13.89

0.4017 1.630

0.5021 2.037

0.1306 0.1981

0.5021 71.43

c) Operation angles glutaconic acid

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depends on the operation time of the spins in both the present ESR-QC systems, resulting in a marked difference from the NMR-QC experimental scheme. 60


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Conclusions In this work, we have for the first time executed the ESR-based QC for the factorization of 21 by an AQC algorithm, implementing the pulse sequences of spin qubit operations and we have estimated how fast the factorization is processed on real molecular spin QCs in the solid state, compared with the previous NMR-QC case. The pulse sequences were implemented by assuming the real experimental conditions and by the restriction only between the hyperfine tensors and static magnetic field for the secular approach as a standard one, although we note that there are other approaches for the pulse ESR technique (e.g. average Hamiltonian method [27,35] etc.). The present approach suggests that molecular spin qubits with spin properties optimized can afford to perform AQC with much faster than NMR-QC cases. ESR-QC experiments at X-band on the glutaconic acid system are underway by using AWG-based spin manipulation technology at liquid helium temperatures. The conquered pulse sequences for the 3e and 1e+2n systems give small required computation times enough to utilize the strong interactions occurring in the adopted molecular spins. The difference in generating a number of pulses between the present approach and the NMR-QC case was identified in terms of the number of the pulses and the spin rotation angles, suggesting any reduction in the number of pulse operations. The pulse intervals in the present AQC algorithm are found to be much small in ESR-QC cases. This is a particular issue intrinsic to AQC, contrasting with standard gate model approaches to QC. In this context, improved MW spin technology is encouraged to develop from the experimental side. Finally, we emphasize that the present approach to perform AQC on molecular spin QCs requires only three spin qubits and such a small number of qubits are not enough to solve intractable problems with CCs. We note that AQC is another approach to QC but equivalent to the standard quantum gate model. All physically realized qubits face their scalability in extending the QC capability [2,12,25,26]. The present work suggests that the scalability is materials challenge and chemistry can reach an important milestone if more than ten addressable synthetic spin qubits are prepared in optimized molecular frames. Based on the possible scenario, we have illustrated how chemistry contributes to the development of QC/QIP in spite of the fact that quantum chemistry or quantum chemical calculations performed on QCs is still a challenge for current QC/QIP technology underlain by quantum gate/circuit models. It is worth mentioning that in this work we have achieved adiabatic quantum computing (AQC) on molecular spin QCs composed of a few addressable qubits and demonstrated that molecular spins can afford to execute realistic quantum computing.

Acknowledgements 55

This work has been supported by Grants-in-Aid for Scientific This journal is © The Royal Society of Chemistry [year]

Research on Innovative Areas "Quantum Cybernetics" and Scientific Research (B) from MEXT, Japan. The support for the present work by the FIRST project on "Quantum Information Processing" from JSPS, Japan and by the AOARD project on “Quantum Properties of Molecular Nanomagnets” (Award No. FA2386-13-1-4030) is also acknowledged.

Notes and references 65

Department of Chemistry and Molecular Materials Science, Graduate School of Science, Osaka City University, 3-3-138, Sugimoto, Sumiyoshi, Osaka 558-8585, Japan. Fax: +81-6605-2522; Tel:+81-6605-2605; E-mail: [email protected], [email protected] † Supplementary Information available:

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Molecular spin QCs for adiabatic quantum computing: A phthalocyanine derivative with three electron qubits and a glutaconic acid radical with one electron bus qubit and two nuclear client qubits. 160x79mm (150 x 150 DPI)



DOI: 10.1039/C4CP04744C

Hybrid spin and valley quantum computing with singlet-triplet qubits.

Characterizing Si:P quantum dot qubits with spin resonance techniques.

Scalable quantum computing based on stationary spin qubits in coupled quantum dots inside double-sided optical microcavities.

Universal quantum gates on electron-spin qubits with quantum dots inside single-side optical microcavities.

High-fidelity gates in quantum dot spin qubits.

Quantifying the quantum gate fidelity of single-atom spin qubits in silicon by randomized benchmarking.

Quantum information processing with nanomechanical qubits.

Quantum computing: powered by magic.

Adiabatic quantum simulation of quantum chemistry.

Experimental implementation of assisted quantum adiabatic passage in a single spin.

Flux qubits with long coherence times for hybrid quantum circuits.

Phonon-mediated generation of quantum correlations between quantum dot qubits.

Error-corrected quantum annealing with hundreds of qubits.

Quantum computing. Defining and detecting quantum speedup.

Bifurcation-based adiabatic quantum computation with a nonlinear oscillator network.

Quantum computing: Three of diamonds.

Two-photon interference at telecom wavelengths for time-bin-encoded single photons from quantum-dot spin qubits.

Manipulating molecules with quantum light.

Exact nonadiabatic holonomic transformations of spin-orbit qubits.

A photonic platform for donor spin qubits in silicon.

Coherent operations and screening in multielectron spin qubits.

Optical Control of Donor Spin Qubits in Silicon.

Quantum correlated heat engine with spin squeezing.

Manipulating quantum information with spin torque.