Two-qubit logical operations in triangular arrangement of quantum dots
We propose a system of two interacting always-on, exchange-only qubits for which , , quantum Fourier transform () and operations can be implemented only in a few electrical pulses in a nanosecond time scale. Each qubit is built of three quantum dots (TQD) in a triangular geometry with three electron spins which are always kept coupled by exchange interactions only. The qubit states are encoded in a doublet subspace and are fully electrically controlled by a voltage applied to gate electrodes. The two qubit quantum gates are realized by short electrical pulses which change the triangular symmetry of TQD and switch on exchange interaction between the qubits. We found an optimal configuration to implement the gate by a single pulse of the order 2.3 ns. Using this gate, in combination with single qubit operations, we searched for optimal conditions to perform the other gates: , and . Our studies take into account environment effects and leakage processes as well. The results suggest that the system can be implemented for fault tolerant quantum computations.
pacs:73.63.Kv, 03.67.-a, 03.65.Xp
The basic unit in quantum computers is a qubit, which can be physically realized in superconducting circuits (1), trapped ions (2), single photons (3), molecular magnets (4), or a single defect in diamonds (5). Recent progress in experimental fabrication of semiconducting quantum dots (QDs) makes them one of the most perspective for quantum computation (6). By standard lithographic methods one can achieve large arrays of QDs which can work as multi qubit quantum register required in universal quantum computations. Full control of the device can be performed purely electrically by gate voltages applied to external electrodes. Sensitive methods of detection single electron dynamics in QD, for example by measurements currents in quantum point contacts (QPC) (7) or in single electron transistors (SET) (8) give the opportunity to read-out the qubit states with very high accuracy.
Loss and DiVincenzo (9) proposed a spin qubit encoded in a single electron spin in a QD which is characterized by longer coherence time than a charge qubit. To encode the spin qubit one needs to applying an external magnetic field which removes the spin degeneracy. The control of the qubit states is performed by the electron spin resonance (ESR) (10). Recently the two-qubit logic gate was performed by Veldhorst et al. (11) in isotopically enriched silicon double quantum dot systems. Another proposition is to encode the qubit in two spin states where a singlet state and one of triplet states (S-T) correspond to the north and the south pole of the Bloch sphere. The qubit rotation around one of the axis on the Bloch sphere can be performed in a nanosecond scale via a pure electrical control of the exchange interaction (12). The rotation around the second axis can be induced by a controlled dynamic nuclear polarization (13) or by a magnetic field difference between two sides of the double dot with integrated micromagnets (14). There are propositions (15); (16) to build a system of two interacting S-T qubits. Mehl et al. (15) proposed the high-fidelity entangling quantum gate in two S-T qubits mediated by one quantum state from a quantum dot between them. The S-T qubits can be also coupled via an exchange (17) and a capacitive interaction (16).
One of the most promising concept is an exchange-only qubit encoded in a doublet subspace of three spins (18). The advantage of this proposal is easy control of the qubit states by purely electrical manipulations of the exchange interactions between the spins. Moreover, the doublet subspace is protected from decoherence processes (19). The exchange-only qubits can be encoded in three quantum dots (TQD) with a linear configuration (20); (21); (22), a triangular arrangement (23); (24); (25); (26) or in a double dot system with many levels (27). Initialization and one-qubit operations are performed by electrical pulses applied to gate electrodes and was already demonstrated experimentally for the linear TQD (l-TQD) (22) and theoretically for the triangular TQD (t-TQD) (25); (26) (for a recent review see (28)). Manipulation of the qubit can be also done by an rf voltage applied to one of gate electrodes in a resonant exchange qubit (29); (30). If the rf excitation energy matches to the energy difference between the qubit states one can observed the Rabi nutation on the Bloch sphere. The read-out of the qubit states can be done by measurement of the current flowing through the system in the doublet (25) or quadruplet (31) blockade regime. Recently an always-on exchange qubit (AEON) (32) was presented in the linear TQD system. In such configuration all exchange couplings are always kept on during the qubit operations, which differs from previous concepts. An advantage of this proposal is performing the quantum logical operations at a sweet spot in detuning parameters, where charge fluctuations are minimized (28); (29); (33).
A very important challenge is implementation of a two-qubit logical operations for which one of the most effective is the controlled phase gate (). This gate in combination with single-qubit gates can be used as a circuit for any universal quantum computation. Recently Doherty and Wardrop (34); (35) showed theoretically how to implement the gate by a single exchange pulse in the resonant exchange qubit encoded in the linear TQD system. They estimated the gate operation time at 21 ns. Pal et al. (36) proposed capacitively-coupled two exchange-only qubits for which the CNOT gate can be performed by varying the level splitting of individual qubits and the inter-qubit coupling time.
In this paper we consider two interacting AEON qubits each encoded in the triangular configuration of TQD. It has been shown that the qubit states are sensitive to breaking of the triangular symmetry (23); (24); (25). Moreover, in the triangular TQD any qubit state on the Bloch sphere can be easily generated by an adiabatic Landau–Zener transition (26), which is in contrast to the linear geometry where one of the pole of the Bloch sphere is favorable. This gives opportunity to construct multi-qubit register where each qubit can be encoded in a desired state. The symmetry of TQD as well as an interqubit coupling can be fully electrical controlled by gating the exchange interactions between the spins. Our main purpose is to study the two-qubit operations as , , quantum Fourier transform () and . First we will consider implementation of the gate in two coupled triangular TQD systems and show that it can be performed by a single electrical pulse only. Next this gate, in combination with the one qubit operations, will be used to search a most optimal configuration to perform the and gates. In the previous paper (25) we showed that one-qubit gates can be performed in a single step by a quick change of the symmetry of the triangular TQD system. This gives the opportunity to implement the and the gates in 3 pulses only. In the earlier paper, by DiVincenzo et al. (18), required 19 pulses and by Shi et al. (27) – 14 pulses. The operation is usually implemented by three gates (37). Moreover we will show how directly perform by only two pulses: switching on the exchange interaction between the qubits and simultaneously performing the Pauli X-gate.
Our research of two-qubit logical operations is supplemented by an analysis of a role of an environment which disturbs the quantum system and its control. In the exchange only qubits, the main sources of the decoherence are the magnetic noise (38) due to nuclear spins and the charge noise (39) related with the random potential fluctuations of the experimental set-up. For the single qubit those effects can be suppressed by encoding the qubit in the DFS subspace (19) and operating in the sweet spot (32). In this paper we focus on potential fluctuations breaking the triangular symmetry of the system and their influence on two-qubit logical operations. We estimate the fidelity and leakage of the two-qubit gates performed close to the optimal conditions. These results are essential for implementation of the considered systems in fault tolerant quantum computations.
Ii Modeling of two-qubit system
We will consider two interacting exchange only spin qubits, each built on three coherently coupled quantum dots (TQD) in the triangular geometry - see figure 1.
ii.1 Single qubit
First, we briefly describe the single TQD system, which dynamics is govern by an extended Hubbard Hamiltonian
where is a local site energy, is a hopping parameter between the dots and describes a intra-dot Coulomb interaction. The direct interaction originates from a quantum exchange term of the Coulomb interaction between electrons on the dots and (40). For a defined confined potential it can be calculated by means of the Heitler–London and Hund–Mulliken method as a function of the interdot distance, the potential barrier and the magnetic field (41); (42). In experiments on exchange qubits these parameters can be purely electrical controlled by potential voltages applied to the quantum dots (21); (22). It has been shown (43) that for several spins engaged in mutual interactions, both the quantitative and qualitative effects arise which modify the standard form of the Heisenberg exchange interaction. The last term in (II.1) corresponds to the Zeeman splitting by an external magnetic field ( is the Bohr magneton, g is the electron g-factor).
We assume that the qubit system is deep in the charge region (1,1,1), with one electron on each dot. For the large Coulomb interaction one can use the canonical transformation (44), which excludes the local two-electron states, to get an effective spin Hamiltonian:
Here, a total exchange interaction contains the direct exchange, , and the Anderson kinetic exchange , which is derived within the second order perturbation theory as .
The Hamiltonian (2) describes the AEON qubit for which the exchange interactions are controlled electrically by the gate potentials applied between two neighbour dots (45). For the linear approximation , where is a sensitivity of the exchange coupling to the gate voltage . This kind of the exchange control can keep the qubit always in the sweet spot which causes less charge fluctuation than e.g. the detuning the local energy levels on each quantum dot (32).
In this paper we are interested in the analysis of the symmetry breaking effects, therefore it is more suitable to express the gate voltages as an effective electric field . For a small value of the exchange couplings can be expressed as
where , is the vector showing the position of the -th quantum dot and is the angle between the vectors of electric field and .
For three spins in single TQD there are two possible subspaces, with the quadruplets and the doublets. The quadruplet states with the total spin and are given by:
and similar functions for opposite spin orientations. Energy of these states is . The second subspace is formed by the doublet states with and . For we choose the basis:
where is the singlet state and , are the triplet states on the bond . Similarly one can express the doublets for reversing all spin orientations.
We assume that the qubit A (B) is encoded in the doublet subspace and , Eqs.(5)-(6). In further considerations the spin index is omitted for simplification of the notation. The Heisenberg Hamiltonian (2) in the qubit basis can be rewritten in the form, where the index A (B) have been added to distinguish the qubits:
where and are the Pauli matrices and
Notice that describes the single qubit in an effective magnetic field , with and to be its the x and z component. The eigenvalues of are:
where is the doublet splitting. Taking parameters suitable for Si/SiGe quantum dots (46) one can estimate eV. Similarly for qubit B.
The qubit initialization can be performed by an adiabatic Landau–Zener transition (21); (26) to the charge region form a neighbour charge state. In this passage one can control, by another set of potential gates, the exchange couplings, and finally, reach the sweet spot with well defined the triangular symmetry of the TQD system, with a given orientation of the electric field . If , the electric field is oriented toward the dot 1, the qubit parameters are and , then the qubit is encoded in the state - pointing the south pole of the Bloch sphere. For the opposite orientation of the electric field, , the parameter which enables preparation of the qubit in the state - pointing the north pole. Notice, that this procedure always provides initialization of the qubit in the ground state.
After the Landau–Zener passage to the sweet spot and the initialization of the qubit one can perform one–qubit quantum gates - only a single step is needed which changes the symmetry of the system, i.e. a change of the angle of the electric field (25). Taking one can perform the rotation of the qubit state around the x axis of the Bloch sphere. For this case the parameters , and from the solution of the Shrödinger equation one can find a unitary operator of the rotation around the x axis as: . The Pauli X-gate can be performed in time for which the qubit state is changed to the opposite one. Similarly, one can make the Pauli Z-gate operation taking for which the qubit state rotates around the z axis and its evolution is govern by the operator . If one changes the angle to the parameters and become equal. It induces the rotation of the qubit state around the vector which corresponds to the Hadamard gate with and the operation time .
The presented model is general and can describe the linear molecule as well. For the linear molecule one of the exchange interactions is zero. In this case the effective electric field cannot be oriented in the full angle, and the adiabatic generation of the qubit is limited to one hemisphere of the Bloch sphere only. For example for , the electric field angle can take the values and the encoded qubit is oriented to the north hemisphere (the parameter and ). To encode the qubit on the south hemisphere one needs to perform a diabatic Landau–Zener transition in which the excited state is involved as well (26). One can also initialize the qubit in the ground state and by proper sequences of pulses perform the Pauli X-gate (47). This method requires at least three additional pulses which increases the operation time.
ii.2 Two interacting qubits
Let us now consider two triangular TQD systems interacting with each other as presented in Fig. 1a. The total Hamiltonian can be expressed as
where the interaction term is confined to two neighbourhood spins in the system A and B and is given by
Here is an exchange coupling parameter which can be controllable by a potential gate applied a tunnel barrier between the qubits.
where corresponds to the state in the qubit A and B. The total spin of two qubit state is with . The total two qubit Hamiltonian can be written as
This Hamiltonian can be generalized for any orientation of the electric fields
where is the rotation matrix for the two qubit system, and describes the rotation of the electric field in the qubit A and B, respectively.
The Hamiltonian (18) is general but rather complex. Therefore, to simplify considerations we assume that the inter-qubit coupling is small compared to the intra-qubit couplings and two qubit dynamics can be well described by a Hamiltonian derived in the lowest order perturbation theory with as a perturbative parameter. Using the Pauli matrix representation the interaction Hamiltonian can be written as
Here all the parameters are functions of the angles and , and they are proportional to . and describes the shift of all two qubits levels and the levels in the single qubits due to the perturbation by . For two qubit logical operations important parameters are and which describe the effective coupling between the qubits in the direction and the plane. Table 1 presents these parameters for some specific angles and , when the electric fields are directed to the dot and in the qubit A and B, respectively. For example corresponds to the Hamiltonian given by (17).
The gate demands , moreover the larger value results a shorter operation time, therefore is the optimal configuration for this gate. The case has non-zero and can be use to implement the gate. The similar table, but for the linear TQD configuration, can be found in (34).
The above analysis has been confined to the cases with a single connection between the qubits, however, one can easily generalized it for a multi-connected TQD system. When two TQD systems are connected by their bases then the interaction Hamiltonian has similar form (II.2) with the parameters being a sum of those from Table 1 for an appropriate direction of the electric field -. For example, for the case presented in Fig. 1b, the interacting Hamiltonian is . We expect that for multi-connections the operation time should be shorter for some two-qubit gates.
Iii Two–qubit quantum gates
Let us study the dynamics of two interacting qubits and show how to perform two-qubit quantum logical operations like , , and . We would like to find most optimal schemes of control pulses which implement these quantum gates. To this end we will consider the evolution of the two-qubit state
derived from the time dependent Schrödinger equation for the Hamiltonian (16) with an appropriate series of pulses.
First, we focus on the gate, because it is one of the most universal gate which in combination with one-qubit gates can be used to perform any quantum algorithm. This gate is defined by the diagonal matrix
where the phase is added to the qubit B (the target qubit) if and only if the qubit A (the control qubit) is in the state . The gate can be performed by a single electrical pulse which switches on the exchange interaction between two qubits. From Table 1 one can see that the simpler implementation of the gate can be done for the electric field orientation - or - (-). In these cases the parameter , and therefore, the Hamiltonian (LABEL:H_eff) has an effective Ising form and the logical gate is provided by the interaction . The operation time can be estimated from the condition (34); (35) and is shortest for the largest . Table 1 shows that the parameter has the largest value for the orientation -, and thus it is the most optimal configuration to perform the gate. Taking , and the exchange parameters eV one can estimate the gate time as ns for Si quantum dots (48).
for in the combination with the Hadamard gates can be used to perform the gate. The operation flips the state on qubit B if and only if the qubit A is in state and is given by
where is the Hadamard gate performed on the qubit B (37). The scheme of the realization of the gate is presented in Fig. 2. In the first step (see Fig. 2a) the system is initialized in the state (for the angle ). Next, a single pulse is applied to the qubit B which changes to and the parameters and become equal. It induces a rotation around the vector on the Bloch sphere which corresponds to the Hadamard gate. Afterwards the interaction between the qubits is switched on and the gate is performed in time , which is seen as the rotation around z-axis over on the Bloch sphere. Finally one needs to apply another Hadamard gate to the qubit B. Notice that only during the operation which reduces leakage processes. Figure 2b presents the occupation probabilities of the two-qubit states during the operation. At the initial time the system is at the state and the Hadamard gate transforms it to the superposition . The operation time is , which can be estimated as ns taking eV for Si/SiGe quantum dots (46). Next the operation transforms the system to the state in the time ns. Notice that the whole operation requires 3 pulses only.
As already mentioned, can be used in any quantum algorithms, e.g. quantum factoring, quantum phase estimation for finding eigenvalues of a unitary operator as well as the order-finding problem. In all of these algorithm a key ingredient is the quantum Fourier transform () (37). is the unitary operation for performing a Fourier transform of quantum mechanical amplitudes. The quantum circuit for the two-qubit can be expressed as
and its realization is presented in Fig. 3. It is similar to the gate, however the first Hadamard gate is performed on the qubit B whereas the second one on the qubit A. Moreover, the gate in between changes the phase of the qubit B by , which means that the operation time is twice shorter. The total time needed for this gate can be estimated as ns.
We would like to consider the operation which can be used e.g. in quantum teleportation (49). The gate swaps the qubit states and can be implemented by three gates according the scheme presented above. However, for (e.g. for the orientation -) the gate can be performed directly in two pulses only. In the initialization step the qubits are encoded in the state by taking , for . Next, the effective electric field is reversed by a single pulse in the qubit B ( is changed to ), which changes the qubit symmetry as well. For two decoupled qubits () this operation rotates the state of qubit B around -axis on the Bloch sphere which is equivalent to the Pauli -gate (25). However, if the interaction between the qubits is switched on by the second pulse () then the state on the qubit A simultaneously rotates around the -axis, which corresponds to the operation. Notice that both pulses are applied at the same time. In this case the solution of the Schrödinger equation is
where the phase factor . One can easily find the operation time , which for the considered orientation - is and is estimated as ns for Si devices (48). The operation time can be even shorter for multi-connected TQD systems. For example, for two triangles connected by their bases the interaction Hamiltonian is the sum , then the parameter is twice larger which implies twice shorter operation time.
Iv Fidelity and Leakage
The considerations above have been performed for the ideal generated qubits, with precisely defined initial states, for the specific angels and of the electric fields and applied in the two-TQD device. However in a real device there are many obstacles to reach these conditions, e.g. an unperfect experimental arrangement or magnetic and charge noises. Since both qubits are encoded in the doublet subspace which is the DFS the system is immune against global magnetic field fluctuations. The other magnetic noise comes from the nuclear spins surrounding the electron trapped in the quantum dot. This local magnetic field, called the Overhauser field, causes leakage out of the computation subspace. To suppress this effect we assumed that both the Zeeman magnetic field and the exchange interaction are larger than the Overhauser field. Because of the electrical control of the exchange-only qubits the charge noise should be taken into consideration as well. It was shown (39) that for a single qubit controlled by gating the potential barriers between the quantum dots the sweet spot is located deep in the (1,1,1) regime, where the estimated dephasing time is of the order of s.
Our studies concern two AEON qubits, where small detuning of the exchange couplings changes the triangular symmetry and all quantum operations are performed in the sweet spot. Our aim is to consider mismatch of the angles and of the electric fields on the gate realization. If the angles are not precisely defined, e.g. caused by fluctuations of the gate potentials, then the initial state can be in some superposition with another qubit state. It has impact on realization of a logical operation and a final two-qubit state. The accuracy of the performed operation can be described by the fidelity which is a measure of the distance between two-quantum states, a desired ideal state and a real final state after the gate operation (37). It can be expressed by
where and are the corresponding density matrices. For the perfect qubit gate the fidelity is unity, however in real devices it is lowered due an imperfect setup of the initial state (the mismatch of and ) as well as leakage processes.
We focus on the gate for which the desired state is defined by
After the operation all the coefficients should be unaffected, , except which changes its sign.
Fig. 4a) presents the fidelity for the operation plotted as a function of the angle (or ) for keeping (or ) fixed. At the parameters and the initial state is precisely. In this case the fidelity is . Its value is close to unity but is not unity, because in the calculations we included leakage processes as well (this issue will be discussed in details later). If the initial state is not precisely generated due to a mismatch of the angle (), then the parameter and the single qubit is encoded in the state with some contribution of the state which disturbs the evolution of the qubit. One can observed a coherent rotation between the states and with the oscillation period of the order of (comparable with ). It affects the real final state and leads to a decrease of the fidelity . When the control qubit A is in state , then the gate does not change the state of the target qubit B. The two qubit system is much less sensitive to the symmetry breaking in both the qubits and the fidelity is close to unity, (see the red curves in Fig. 4a).
Now we discuss in details the leakage which is a measure of how much of the initial qubit state diffuses out of the logical qubit basis. For two interacting TQD systems the total Hamiltonian , Eq.(13), conserves the total spin as well as its z-component . The leakage from the two-qubit space (15) is related with the interaction Hamiltonian , which does not preserve the local spin numbers in the qubits A and B. There are 11 leakage states, which can be constructed from the quadruplets and the doublets , as: , , , , etc.
The leakage can be defined as , where is the projector off the computation subspace (35). The right plot in Fig. 4 presents after the operation for the orientation 1-1. In the calculations we used the time dependent Shrödinger equation with the Hamiltonian (13) and all the states with the total spin and . We found that the state is coupled to 7 leakage states. The largest contribution is due to the state for which is the same order as those ones in the table 1. However, the energy gap between the qubit states and all leakage states is very large, , therefore the spin flip which results from is energetically unfavorable. This leads to the leakage , see Fig. 4b). For the initial state the leakage , because this state is not coupled to the leakage states. These results suggest that the considered system can be implement for fault tolerant computations (50).
We have shown how to perform , , and operations in two AEON qubits each encoded in three coherently coupled quantum dots (TQD) in the triangular geometry. Each of the qubit consists three electrons and the qubit states are encoded in the doublet subspace which is sensitive to the triangular symmetry breaking. This subspace was pointed out as the decoherence-free subspace (19) which is immune against magnetic noises. The symmetry of the system is fully controlled by short voltage pulses (in a scale of nanoseconds) which are applied between the quantum dots to change exchange couplings. Notice that each qubit operates in the sweet spot, deep in the (1,1,1) charge region for which the local charge noise on the quantum dots can be neglected. An advantage of the triangular geometry is simpler generation and encoding both of the qubit states by adiabatic Landau–Zener transitions which is in contrast to the linear geometry where one state is preferable. For various orientations of the electric fields in both TQD systems we have shown the most optimal configuration for performing the two-qubit operations.
The gate can be implement easily by means of a single electrical pulse which switch-on the interaction between the qubits. The most optimal configuration for realization this gate is the configuration of the electric field oriented to the dot 1 in both the qubits (see Fig.1) and the estimated operation time for Si quantum dots is ns at . The gate, in combination with two Hadamard gates, are used to performed gate. This gate requires 3 pulses only: the first and third one changes the symmetry of the second qubit and they are used to perform the Hadamard gates, whereas the second pulse performs the gate. It is an advantage compared with the linear system for which one needs 19 pulses to realize the gate (18). A very similar sequence of pulses can be used to perform the quantum Fourier transform, here – changes the phase of the target qubit by , and the first Hadamard gate is applied to the qubit B whereas the second one to the qubit A.
We also showed how to perform the gate. This operation requires the non-zero coupling between the qubits in the - plane (with the parameter ) and the optimal initial configuration is for the orientation -. This operation needs only two pulses and time ns.
Moreover, we considered the fidelity of the two-qubit operations which for the is very large and calculated at the initial state and , respectively. When the control of the symmetry of the system is disturbed by environment effects the fidelity is reduced but still very high. We estimated the leakage during the gate operation which is very small, the order of . Since a threshold for quantum computations was estimated at the level (50), our studies show that TQD systems can be implemented for fault tolerant computations even without applying noise corrections sequences (51).
Acknowledgements.This work has been supported by the National Science Centre, Poland, under the project 2016/21/B/ST3/02160.
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