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Quantum Computation with Magnetic Clusters

We propose a complete, quantitative quantum computing system which satisfies the five DiVincenzo criteria. The model is based on magnetic clusters with uniaxial anisotropy, where standard, two-state qubits are formed utilizing the two lowest-lying states of an anisotropic potential energy. We outline the quantum dynamics required by quantum computing for single qubit structures, and then define a novel measurement scheme in which qubit sates can be measured by sharp changes in current as voltage across the cluster is varied. We then extend the single qubit description to multiple qubit interactions, facilitated specifically by a new entanglement method which propagates the controlled-NOT (C-NOT) quantum gate.


Introduction
In classical, digital computation, information is processed and stored in two distinct states which are usually referred to as 0 and 1. A classical computer processes one input at a time to calculate the output. On the other hand, in quantum computing (QC), one can make use of quantum superpositions to explore multiple inputs and corresponding outputs simultaneously 1,2 . This feature gives QC important advantages for certain tasks such as searching for an item in an unsorted database. Certain quantum sorting algorithms can complete such a query in √ tries whereas classical algorithms would require on the order of tries, where is the total number of items in the database 3 . Additionally, quantum computers can handle problems like the primefactorization of large numbers 4 which are practically impossible even for supercomputers.
The basic element of a classical computer is a bit, which can store 0 or 1. In a quantum computer, the fundamental element is called a qubit, which is a physical system with two well-defined states, represented by |0〉 and |1〉. The crucial difference between a qubit and a bit is that a qubit can be in a superposition of states, i.e.
where | 〉 is the wave function describing the state, and and are arbitrary coefficients having the property 1. Quantum algorithms involve operations on individual qubit states as well as operations that require inter-qubit interactions. A multi-qubit quantum computer must be able to create entangled states by exploiting physical interactions between qubits, which can be realized as Heisenberg type spin-spin coupling, spin-orbital coupling and several other types of interactions. The physical nature of the qubits and the interactions depends on the physical system comprising the quantum computer.
There have been several different proposals for physical implementation of qubits; for instance, some approaches use Josephson Junctions as qubits 5,6,7,8,9,10 . There are also methods using the polarization and propagation direction of photons as the states of the qubits 11 . Nuclear magnetic resonance QC 12,13,14,15 (NMR QC) uses the nuclei of certain molecules to form qubits. A macroscopic number (10 19 ) of these molecules are dissolved in a liquid and placed in a strong bias magnetic field (~10 T) along the z-axis. The interaction of the nuclear magnetic dipole with the applied field gives two, well-defined states, spin up and down, represented by |0〉 and |1〉 respectively. Furthermore, a small, time-varying magnetic field is applied along the x-axis. The phase and frequency of this magnetic field are precisely adjusted to control the dynamics of individual types of nuclei in the sample to create single-qubit operations. In addition to the interaction with the external field, qubit dynamics is governed by inter-qubit, Heisenberg-type interactions 16 . Making use of the inter-qubit interactions and single-qubit operations, one can implement the controlled-NOT (C-NOT) operation, and create entangled states 15   can be generated using an oscillating current through the top and bottom contacts and a switch (not shown) which allows for single qubit addressing.
The system is placed in a large magnetic field which is aligned with the anisotropy axis of the MC. Furthermore, a small, oscillating magnetic field is applied in the perpendicular direction to control the dynamics of the qubit. The state of the qubit can be read out by applying a voltage across the terminals, the details of which will be discussed in the Readout section. Before we start discussing the details of the MC based QC, we compare its properties with several available QC approaches, summarized in Table 1

Quantum Description of Magnetic Cluster
The MC, depicted in Fig. 1, has a uniaxial magnetic anisotropy along the axis of the applied magnetic field, . When the transverse magnetic field, , is turned off, the Hamiltonian that governs the dynamics of this system can be written as: , where is the anisotropy energy, is the spin operator, is the gyromagnetic ratio, and is  After the initialization, which will be discussed in the Read out and Initialization section, the qubit will be in the ground state, |0 , which corresponds to . The first excited state of the qubit is the one corresponding to 1, denoted by |1 . The energy of each level and the gaps in between are given in Table 2. The energy required to make a transition from the first excited state to the second is ∆ ≡ 2 3 . Transitions between states are activated by the absorption and emission of phonons 30,31 . If the temperature is such that ≪ ∆ , where is the Boltzmann constant, the number of phonons with enough energy to make such a transition is exponentially suppressed. Therefore, we can limit the discussion to the subspace spanned by |0 and |1 , where |0〉 ≡ 1 0 and |1〉 ≡ 0 1 . In this subspace, can be represented by a 2x2 matrix, which can be rewritten in terms of as follows:

State Energy Excitation Energy
In this representation, we can rewrite the Hamiltonian as: where 2 1 and is a constant which is dropped from the Hamiltonian.
Having described the Hamiltonian of the system, we now proceed to discuss how single-qubit operations may be implemented.

Single Qubit Operations
Now, we introduce a time dependent magnetic field of magnitude along the x-axis to generate single-qubit operations. With this magnetic field, the effective Hamiltonian becomes: , where ω γB ≪ , is the frequency of the control magnetic field (on the order of GHz), and is the initial phase. Since the Hamiltonian is time dependent, it is convenient to transform the wave function into the so-called rotating frame, Inserting equation (6) into the Schrodinger Equation yields which shows that the Hamiltonian in the rotating frame is where we define Σ ≡ and Δ ≡ . Because is the frequency of the control field, we can choose it such that , which gives Σ 2 and Δ 0.
Furthermore, the terms with Σ are rapidly oscillating, and their average becomes zero over the time scale, 1 ⁄ , which is the time scale for rotations. This approximation is called the rotating wave approximation 16 . In this limit becomes time independent, and reads .
The time development operator at ⁄ can be written as which can be used to generate rotations around the x and y axes. Although equation (10) lacks , the rotations around the z-axis in the subspace spanned by |0 and |1 can be generated as a series of rotations around the x and y axes. This follows from the identity Therefore, we conclude that | 〉 can be rotated to any point in the Bloch Sphere, which means any superposition of |0 and |1 can be realized. Now that we have shown single-qubit operations are possible with MC QC, we will turn to the description of how a qubit is initialized and how the final state is read.

Initialization and Read Out
Qubits of a quantum computer must be initialized before performing an algorithm. After initialization, quantum algorithms require a series of single-qubit and multi-qubit operations. A readout method is required to determine the result of the computation.   voltage is applied such that ≫ . This will ensure that the LUMO will be unoccupied, and the MC can effectively be described by the Hamiltonian equation (2); therefore, single-qubit operations can be implemented as discussed in the previous section. At the end of the calculation, results are measured in the readout phase.

Read Out
To begin a qubit measurement, the voltage is slowly lowered from levels ≫ maintained during the calculation phase. As seen in Fig. 3, the energy levels of first two states intersect with energy of 1, ⊗ | , therefore transitions from n =0 state to n=1 state are possible. Such a transition is possible by tunneling of an electron from one of the terminals to the MC, which will result in an abrupt change in the current. The value of the voltage at which such a change is observed will give the information on the state of the qubit. Changes in current as is varied can be seen in a differential conductance measurement, where peaks in | / | correspond to state transitions. A transition peak observed at would correspond to the collapse of the state to the first excited state, whereas a transition at ≡ would correspond to the collapse to the ground state. Electrons traveling from one contact plate to the opposite through the MC cause a current which has been calculated to be on the order of pA 27,28 . Because measuring pA currents through the device may pose challenges, an actual implementation of an MC qubit will likely entail an ensemble of MCs which will allow larger currents. This ensemble approach to realizing MC qubits will be discussed in the Ensemble MC QC section. Not only is measurement the key to finalizing a quantum algorithm, but it is also useful in preparing an initial qubit state from which an algorithm can begin.

Initialization
The initial state of a quantum computer can be in a superposition of all possible states. To begin a quantum algorithm, the system must be initialized to a known state, which can be done by letting the system relax to its ground state. If the MC qubit is given enough time to relax, the probability of finding it in a state of energy is approximately The ratio of occupation probabilities for the first excited state and ground state can be estimated as ∆ , which quickly converges to zero if ≪ ∆ . Thus, if the system is kept at low enough temperature, qubits will relax into their ground states, which will be used as the initial states of the QC. Furthermore, keeping ≫ , ensures that the ground state is |0 ⊗ | .

2-Qubit System and Entanglement
Let us consider the 2-qubit system illustrated in Fig. 4. In this setup, the MCs interact with each other by Heisenberg-type coupling. The Hamiltonian for this 2-qubit system is given by: where indices label qubits, and is the interaction strength between the qubits. Following the idea for a single qubit, we transform the wave function into the rotating frame by where and are the Larmor frequencies of the first and second qubit, and | 〉 and | 〉 are the wave functions of the two-qubit system in the lab frame and rotating frame, respectively. If ≫ 1, we can use the rotating wave approximation to calculate the transformed interaction Hamiltonian, which reads In order to create entanglement, we set 0, in other words turn off the control magnetic field, and utilize the last term in equation (16). Let us choose the basis states as where ≡ , and the overall phase can be neglected. The single-qubit operations in equation (18) are defined in equations (10)(11), where the subscript denotes the qubit the operators act on. The resultant operation in equation (18) is the C-NOT operation, which can be used to entangle qubits. The description developed for the two-qubit system can easily be generalized to multiple qubits which is done in the next section.

Multi-Qubit System
The Hamiltonian for a multi-qubit system can be written as where is the coupling strength between the i th and j th qubit. The coupling strength, , may decay quickly with distance separating qubits which makes it difficult to directly entangle two nonadjacent qubits. However, one can still entangle any two qubits by a series of operations on subsequent neighboring qubits. For instance, let us consider a three-qubit system for which there is no direct interaction between qubit 1 and qubit 3 (i.e. 0). In this case, one can create a C-NOT operation between the first and third qubit as follows: where the C-NOT operations on the right hand side of the equation are implemented by the couplings between neighboring qubits. This idea can easily be generalized to any multi-qubit system.

Decoherence
Qubits are physical systems coupled to their environment which can cause the qubits to lose their quantum nature. The decoherence time of a quantum computer is the lifetime of coherent quantum states. In the case of MC QC, the interaction of the MC with the crystal structure and phonons may result in a transition out of the two-state subspace previously defined. In this section, we discuss the dynamics of decoherence, and calculate the lifetime of an MC qubit, focusing on decoherence originating from phonon-assisted transitions out of the qubit's two-state subspace. We turn to references [29,30,31] for the detailed description of the spin-phonon interaction. To begin, we must consider physically what occurs in a single-state excitation facilitated by absorption or emission of a phonon, which is depicted in Fig. 5.

Figure 5: Phonon Stimulated Excitations
The states of the system, a few of which are shown in Fig. 5, have different energy and angular momentum values; therefore, any transition between two states requires angular momentum and energy transfer to or from the qubit. Excitations in the crystal structure will carry in (or out) the required angular momentum and energy difference. These excitations can be described as transverse phonons, denoted as . Phonons cause perturbations on the angles of the crystal axes, which can be written as .
We can calculate the Spin-Phonon interaction by perturbing the magnetic anisotropy Hamiltonian with the angles as follows: where high orders in are neglected because ≪ 1, and the Hamiltonian, , is the anisotropy contribution of the Hamiltonian defined in equation (4).
The decoherence rate corresponding to the transitions from the subspace to outside the subspace can be calculated utilizing equation (22). This is accomplished by calculating the amplitude for the spin-phonon scattering. We define quantum states, Ψ , , ⊗ , , where the indices, and , refer to the final and initial states. , and , are the eigenstates of the spin and phonon Hamiltonians respectively. Because two adjacent phonon states differ by one phonon quanta, we define the phonon states, where / is the phonon number density, is the number of cells in the crystal structure, is the unit cell volume, is the mass of the cells, and , is the phonon frequency. , the speed of phonons, can be estimated as , where is the lattice constant.
The decoherence time, ≡ Γ , must be long enough such that a large number of single-qubit and multi-qubit operations can be executed before the quantum states decohere. The spin matrix element component for Γ , namely | | ∝ , which can be used to write a proportionality expression for the decoherence time, is proportional to the moment of inertia of the unit cell. With a small enough value and properly chosen MC, the decoherence may be made long enough to perform a useful number of operations. The number of operations allowed by a certain decoherence time can be found by comparing the gate operation time, ≡ , to the decoherence time in equation (24). Using 0.6 / where is Tesla, the application of a magnetic field around 0.01 yields a gate operation time, 0.05 0.5/ . We compare several cases of this decoherence time in Fig. 6 below, where we have assumed an anisotropy energy, 0.1 ,

10
, and gap energy 2 . While Fig. 6 suggests that for MCs with relatively small spin number, 10, decoherence effects are prominent at large temperatures, with sufficient temperature reduction and by choosing an MC with a good spin number, decoherence effects can be successfully suppressed.

Ensemble MC QC
In order to circumvent challenges of measuring the current through a single MC, an ensemble of MCs can be placed between the contact plates to increase the current. Fig. 7 shows a singlequbit ensemble of MCs, placed in a two dimensional array between the contacts. The distance between each MC is such that interactions between any pair is negligible, which allows us to describe the dynamics of each MC individually.
The MCs are placed a distance apart such that MC-MC interactions are negligible within the qubit structure. Because these interactions are negligible, the system can be described by the density of states for the ensemble. This approach is in great analogy with NMR QC, for which ensemble computation techniques are well developed 12 . In order to create a second qubit, another layer of MCs are placed directly above the first qubit such that MC grids are precisely aligned. Therefore, each individual MC in the first layer interacts with only one MC in the second layer. This idea can be extended to the case of multiple qubits by building more layers.

Conclusion
We have provided a theoretical framework for quantum computing using magnetic clusters (MCs) as qubits. With this physical system, we can satisfy all five prerequisites for implementing a quantum computer as follows: 1. The MC qubit is realized in MC spin states which are well-separated energetically, given proper parameters. This energy scales approximately as 2 , where is the spin of the MC, is the anisotropy energy, and is the applied magnetic field.
2. MC qubits can be initialized via cooling to their ground state due to their asymmetric anisotropy energy structure.
3. A useful number of quantum gate operations can be completed within the decoherence time of the quantum computer.

Any two qubits can be entangled via Heisenberg-type interactions and C-NOT
propagation. Thus, a Universal set of quantum gate operations can be realized.
5. Qubit-specific measurements can be performed to probe the quantum state using electrons injected into the lowest unoccupied molecular orbital (LUMO).
The MC qubits are scalable because they occupy a solid state environment which provides ease of fabrication and because any two qubits can be entangled despite a lack of direct proximity.
With novel methods for qubit readout, and non-local, qubit-qubit entanglement, we have strengthened the possibility for creating a quantum computer using magnetic components, and have provided an alternative to the currently explored quantum computing proposals.
Technical difficulties associated with the measurement of small currents can potentially be overcome by using an ensemble of non-interacting qubits, as briefly discussed in the Ensemble MC QC section. This approach is beyond the scope of the current note, and will be addressed in our following studies.