Simple implementation of high fidelity controlled-$i$SWAP gates and quantum circuit exponentiation of non-Hermitian gates

The $i$swap gate is an entangling swapping gate where the qubits obtain a phase of $i$ if the state of the qubits is swapped. Here we present a simple implementation of the controlled-$i$swap gate. The gate can be implemented with several controls and works autonomously. The gate time is independent of the number of controls, and we find high fidelities for any number of controls. We discuss an implementation of the gates using superconducting circuits, however, the ideas presented in this paper are not limited to such implementations. An exponentiation of quantum gates is desired in some quantum information schemes and we therefore also present a quantum circuit for probabilistic exponentiating the iswap gate and other non-Hermitian gates.

The iswap gate is an entangling swapping gate where the qubits obtain a phase of i if the state of the qubits is swapped. Here we present a simple implementation of the controlled-iswap gate. The gate can be implemented with several controls and works autonomously. The gate time is independent of the number of controls, and we find high fidelities for any number of controls. We discuss an implementation of the gates using superconducting circuits, however, the ideas presented in this paper are not limited to such implementations. An exponentiation of quantum gates is desired in some quantum information schemes and we therefore also present a quantum circuit for probabilistic exponentiating the iswap gate and other non-Hermitian gates.
Equivalent (in the sense that they both constitute a universal set of gates together with the set of one-qubit operations) to the cnot gate is the iswap gate which we denoteŜ = |00 00| ± i(|10 01| + |01 10|) + |11 11|. The iswap gate is a perfect entangling version of the swap gate, which is why it is equivalent to the cnot gate. However, the iswap gate has the advantage over the cnot gate that it occurs naturally in systems with XY -interaction or Heisenberg models, such as solid state systems [17,18] and superconducting circuits [19]. Other implementations of the iswap gate include linear optics [20,21] and nuclear spin using qudits [22].
Here we present a simple implementation of a multiqubit controlled-iswap gate which we call c n iswap , where the n indicates the number of control qubits. For a single control qubit this is essentially an iFredkin gate, i.e., a Fredkin gate with a phase of i on the swapping part. The implementation is based using the control qubits to * stig@phys.au.dk † zinner@phys.au.dk tune the target qubits in and out of resonance by following the approach presented in Refs. [38,39], and can be realized using different schemes for quantum information processing. We include circuit design for an implementation of the ciswap gate in superconducting circuits as well as for the c 2 iswap gate in the appendix. The gate is autonomous in the sense that it does not require any outside driving, and the gate time is thus independent of the number of control qubits. When neglecting the decoherence of the qubits we find a fidelity above 0.998 for one control qubit. When including decoherence of the qubits the fidelity stays above 0.99 for up to four control qubits.
Being able to exponentiate quantum gates can be useful in different quantum information schemes such as in continuous variable (CV) systems [40], where exponentiated gates, such as exp(iθX), can be used to operate on the systems [41,42]. Another scheme which might benefit from being able to exponentiate non-Hermitian quantum gates is quantum random walks [43], where non-unitary operations is needed for, e.g., graph coloring [44,45]. We therefore present a quantum circuit for probabilistic exponentiating of non-Hermitian operators, based on the method by [46] which works for exponentiating Hermitian operators. Our method is exact for cyclic operator, i.e., operators fulfillingT n = 1, while it is approximate for all other non-Hermitian operators.
The paper is organized as follows: In Section II we present a simple Hamiltonian and show how it yields an c n iswap gate. We discuss the effectiveness of the gate exploring the single qubit controlled-iswap gate as an example in Section II A. We further, in Section III, present an implementation using superconducting circuits of the ciswap gate and discuss how to expand it to more controls. In Section IV we show how to expand the implementation of the controlled-iswap gate into controlling swapping of an array of qubits. In Section V we present a quantum circuit for probabilistic exponentiating cyclic quantum gates, and discuss its range of validity. In Section VI we provide a summary and outlook for future work.
With the condition that J z J x both terms of H rot will rotate rapidly, and can thus be neglected using the rotating wave approximation, unless all of the control qubits are in the state |1 . The means that the Hamiltonian effectively becomeŝ where subscript C denotes the state of the control qubits, i.e. the first n qubits, and T denotes the state of the target qubit, i.e., qubit T 1 and T 2. The state |1 C = |11 . . . 1 C denotes the state where all control qubits are in the state |1 . We can calculate the time evolution operator by taking the matrix exponential,Û (t) = exp(iĤ rot t), which yieldŝ whereĨ C denotes the reduced identity of the control qubits where the states |1 1 | C have been removed. The identity of the target qubits is denotedÎ T . From Eq. (7) we see that for times T = (2m + 1)π/2J x , m ∈ Z the time evolution operator takes the form of a controlled-iswap gate.
whereŜ T is the two-qubit iswap gate on the target qubits, which swaps the target qubit with a phase of ±i.
The phase on the target qubit depends on the sign of ∓|J x |. For completeness we note that for times T = (2m + 1)π/4J x we obtain the controlled-√ iswap gate [47].

A. Example: The single controlled-iswap gate
In order to illuminate the performance of the system worked as a c n iswap gate we explore the example of the single controlled-iswap gate. We chose this example since not only is it the simplest non-trivial example, it is also closely related to the Fredkin gate. A schematic presentation of the model yielding the controlled-iswap gate can be seen in Fig. 3(a), which corresponds to Eq. (1) with n = 1.
We characterize the performance of the gate by calculating the average process fidelity, which is defined as [1,[48][49][50] where integration is performed over the subspace of all possible initial states and E is the quantum map realized by our system. We simulate the system using the Lindblad Master equation and the interaction Hamiltonian of Eq. (3) using the QuTiP Python toolbox [51]. The result is then transformed into the frame rotating with the diagonal of the Hamiltonian, and then the average fidelity is calculated. For all simulations we have J z /2π = 50 MHz, while we change the transversal coupling, J x /2π, from 5 to 25 MHz. The average fidelity of the simulation can be seen in Fig. 1 together with the gate time. The figure shows both the average fidelity without any decoherence and with a decoherence time of T 1 = T 2 = 30 µs [52]. Without any decoherence we find that the average fidelity increases asymptotically towards 1 as the driving decreases, with the only expense being an increase in gate time. Since decoherence increases over time, a longer gate time means lower fidelity, which is exactly what we observe when including decoherence in the simulations. In this case we find that the fidelity peaks at ∼ 0.995 around J z /J x ∼ 4, which yields a gate time of T ∼ 25 ns. However, we note that the fidelity are dependent on the parameters J x and J z thus changing these will change the fidelity. We also see that for just J z = 2J x we obtain an average fidelity above 0.99 for a gate time T ∼ 15 ns. The oscillation of the average fidelity is due to a small mismatch in the phase of the evolved state compared to the desired matrix in Eq. (7), which disappears when J z /J ∈ Z.
We simulate the C n iswap gate for different n in the optimal ratio between couplings, J z /J x ∼ 4. The result of this simulation is seen in Fig. 2. We observe that the fidelity stays above 0.998 for up to n = 4 control qubits when decoherence is not included. The reason for this is that for larger n the gate resembles the identity more. This is due to the fact that the identity operation is applied to the control qubits, meaning that for a large number of control qubits, the gate will perform the identity on the control qubits and the swapping operation will only be performed on the target qubits. When decoherence is included the average fidelity decreases for larger n as it should, however, we still find a fidelity above 0.99 for up to 4 controls.

III. EXPERIMENTAL IMPLEMENTATIONS
A possible implementation of the controlled-iswap gate using superconducting circuits can be seen in Fig. 3(b). The circuit consists of three transmon qubits [53,54], where two of them are connected by a capacitance and   the third qubit is connected to the other two by Josephson junctions, with as small a parasitic capacitance as possible. By using the diabatic gate approach in Ref. [55] one can minimize the leakage of the capacitive coupling between the two target qubits. Such a circuit has the Hamiltonian where ϕ i are the node fluxes and p T = (p T 1 , p T 2 , p 1 ) are the conjugate momenta. The Hamiltonian is the general Hamiltonian for a c n iswap gate, and in the case of Fig. 3(b) one needs n = 1. As the capacitive couplings yields transversal XX-couplings when truncating to a Ising-type model, we are not interested in the capacitive couplings between the control qubit and the target qubits, and thus we require C z C i , C T i which will leave the capacitance matrix being approximately diagonal, with the exception of the desired capacitance between the target qubits. This leaves only longitudinal ZZ-couplings between the control and target qubits. This limit where the longitudinal coupling dominates over the transversal couplings is within experimental reach [56]. An other way of reaching high-contrast ZZ-couplings could be to use a combination of transmon and flux qubit, and then engineering opposite sign anharmonicities as in Ref. [57]. When truncating the Hamiltonian in Eq. (10) to a two level system one reaches the Hamiltonian in Eq. (3). A detailed calculation going from the circuit design to the gate Hamiltonian can be found in Appendix A together with an example of an implementation of the c 2 iswap gate.

IV. CONTROLLED SWAPPING ARRAYS
Suppose we have multiple qubits which we want to swap in a controlled way, i.e., first swapping two qubits, then swapping two other qubits, and so on. This might be useful in a range of quantum algorithms.
In this appendix we discuss how to expand the idea of the controlled-iswap gate in the main text into a system where we can swap qubits in an array arbitrarily. We will discuss this for the case of an array of first three qubits and then briefly for four qubits, but the ideas will be easily expandable to more qubits.
As a first attempt to create such a system we connect all qubits which we wish to be able to swap to each other with transversal coupling, J x , each of these n qubits are detuned from the average frequency of the qubits, such that ∆ i = ∆ j for i = j = 1, 2, . . . , n. Following the idea of Fig. 3(a) we add a control qubit for each transversal coupling, and couple it with Ising couplings, J z i to each qubit. A schematic representation of the model for n = 3 can be seen in Fig. 4(a). The Hamiltonian for such a system becomeŝ where Ω is the average over all the target qubits frequency, ∆ i is the detuning of the ith target qubit from the average frequency of the target qubits, and the subscript T i indicates the ith target qubit, while the subscript Ci indicates the ith control qubit. If we require that the Ising couplings have the strengths J z i = −∆ i , and require that J z i J x for all i, then at times T = (2m + 1)π/(2J x ), m ∈ Z the time evolution operator for the n = 3 case takes the form The top three ancilla qubits controls the swapping and corresponds to the blue spheres, while the lower three qubits corresponds to the green spheres. The filled circles indicates that the ancilla qubits must be the the state |1 for the swap to be activated, while the non-filled circles indicates that the ancilla qubits must be in the state |1 ; This corresponds to the time evolution operators in Eqs. (12) and (14).
whereĨ C denotes the reduced identity of the control qubits where the states |100 100| C , |010 010| C , and |001 001| C have been removed. The identity of the three target qubits is denoted I T , andŜ ij is the two-qubit iswap gate which swaps the state of the qubits i and j. The quantum circuit of the model can be seen in Fig. 4(b).
From the time evolution operator in Eq. (12) we see that we have complete control over which qubits we wish to swap, depending on the three ancilla qubits, i.e., if we wish to swap qubit i and j we must have qubit Cij in the |1 state and the remaining control qubits in the |0 state, and so fourth. This means that we need an ancilla qubit for each possible swap. If we require all-to-all swapping with n qubits, we would need N = n(n − 1)/2 ancilla qubits in order to control all couplings.
In order to bring the number of ancilla qubits down we only couple one ancilla qubit to each qubit in the array.
The Hamiltonian then takes the form A schematic representation of the model for n = 3 can be seen in Fig. 4(c). If we require that the Ising couplings have the strengths J z i = −∆ i , and require that J z i J x for all i, then at times T = (2m + 1)π/(2J x ), m ∈ Z the time evolution operator for the n = 3 case takes the form Again we obtain full control over swapping of the target qubits, however, this time we need control qubits Ci and Cj to be in the |1 state and remaining control qubits to be in the state |0 , in which case with the ±iswap -operatorsŜ ij swaps the state of the two qubits i and j. We note that we also obtain a three-way swapping operator when all control qubits are in the |1 state. In its matrix representation the three-way swap-operator is an 8 × 8 matrix and takes the form where the two operatorsŜ 1 andŜ 2 are 3 × 3 matrices and operate on the three dimensional subspaces of one and two excitation number, of the target subspace, respectively. In their matrix representation these take the same form which can be used to entangle all three qubits. We consider the special case of T = mπ/3J x , m ∈ Z, for which the operator takes the form This operator can be used to create state belonging to the same non-biseparable classes of three-qubit states as the W state [58].
In Fig. 5(a) we show the model for a four qubit swapping array with all-to-all couplings corresponding to Hamiltonian in Eq. (13) with n = 4. In Fig. 5(b) we present the corresponding gate of the model coming from making the time evolution operator from the Hamiltonian. As above we obtain fully controllable two-qubit swapping between all of the four qubits. We further obtain four three-qubit entangling gates, similar to the one in Eq. (16) and one single four-qubit entangling gate.
In order to test the viability of our analysis we simulate the Hamiltonian in Eq. (13) using the Python toolbox QuTiP using the same approach as in Section II A. Using parameters J z i /(2π) ∈ {−20, 20, 60}MHz and J x = min i |J z i |/5 we find a fidelity of 0.993 at time T = π/(2J x ) = 62.5 ns without including decoherence, and a fidelity of 0.98 when including a decoherence time of T 1 = T 2 = 30 µs.

V. PROBABILISTIC EXPONENTIATING OF CYCLIC NON-HERMITIAN QUANTUM GATES
In this section we present an exact probabilistic method for exponentiating cyclic non-Hermitian gates using an explicit quantum circuit. While our method is exact for cyclic operators it is approximate for non-cyclic operators. The controlled-iswap gate presented in this paper is in fact a cyclic non-Hermitian gate. Note that exponentiating non-Hermitian gates leads to non-unitary gates.
Unitary Hermitian gates can be exponentiated using the method developed by Marvian and Lloyd [46]. Albeit they only present their method for the controlled-swap gate, it works for all unitary Hermitian gates. Here we extend their method in order to exponentiate non-Hermitian gates. Our method is exact for a gate,T , for whicĥ T n = 1 for n ∈ Z and approximately correct if this is not the case. We call gates whereT n = 1 for cyclic gates with cyclic order n. For n > 2 all cyclic gates become non-Hermitian, due to the fact that all eigenvalues of Hermitian matrices must be real and a diagonal matrix D fulfilling the Spectral theorem such thatT =ÛDÛ −1 , whereÛ is a unitary, must then fulfillD n = 1.
Our result become interesting as soon as you want to exponentiate some sort of phase gate, with a phase other than −1, in which case the gate becomes non-Hermitian. This means that the result of such exponentiating will be non-unitary for n > 2. In Table I we mention a few often used non-Hermitian gates and their cyclic order.  Table I. Common non-Hermitian quantum gates and their cyclic order n. Note that we assume φ to be π divided by an integer. The controlled version of the gates mentioned in this table are also non-Hermitian with the same cyclic order. Gate We note that in order to use our method we must be able to perform a controlled version of the gate we wish to exponentiate, i.e., if we wish to exponentiate an iswap we would need a controlled-iswap , as discussed above.
Suppose we have a controlled cyclic gateT working on an arbitrary number of qubits. In order to create a circuit for exponentiating such an operator we must first Taylor Figure 6. Quantum circuit used to exponentiate a matrixT for whichT n = 1. On top we have n − 1 ancilla qubits which are prepared in the state |φ . Each acts as a conditional for aT operation, and finally they are all measured in the {|± }-basis. Note that theT operation does not have to be a two qubit operation, it can operate on m qubits. expand the exponential In total this yields n Taylor terms. This means that our quantum circuit would need n − 1 ancilla qubits to perform the controls. We then apply the controlled gate n − 1 times, each time controlled be a different ancilla qubit. The quantum circuit can be seen in Fig. 6.
We must now prepare the ancilla qubits in the state where N is a normalization which depends on θ, and the state |k indicates a state with k excitations, i.e. we have |0 = |00 · · · 00 , and |1 = |10 · · · 00 , |1 = |01 · · · 00 , or |1 = |00 · · · 01 , etc. Let |γ be the initial state of the target qubits. If we act with the n − 1 controlled-T gates on the initial state |φ |γ , as in Fig. 6 we arrive at the state If we measure the n − 1 ancillae in the {|± } = {(|0 + |1 )/ √ 2} basis, there is a probability of around 1/2 n−1 that we measure |+ in all of the ancillae, if we require θ to be small. This means that the total state becomes which is the desired result. If this state is not measured the experiment must be repeated until the desired result is obtained. We note that if the gate is not cyclic our method works approximately as long as θ is small, in which case the first terms of the Taylor expansion will dominate. This means that we can chose the number of terms we want in our Taylor expansion as the number of ancillae we include in our quantum circuit.

A. Example
For an example of the Hermitian n = 2 case see Ref. [46]. Here we consider the case n = 4. This could, for example, be a controlled-iswap . The exponential in this case becomes Remember that the operator in the exponent is not Hermitian, and thus we are not dealing with a unitary. This means that if θ becomes large, then the hyperbolic functions will blow up. Therefore we keep θ small. Notwithstanding, we prepare three ancillae in the state |φ = N 2 (cos θ + cosh θ)|0 + i(sin θ + sinh θ)|1 All normalization is included in N . Note that we could have chosen other states such as |100 and |101 in the second and third term of |φ as well, as this choice can be made without loss of generality. Now preforming the three controlled T -gates on the qubits we arrive at the state |φ |γ → N 2 (cos θ + cosh θ)|0 + i(sin θ + sinh θ)|1 T + (cos θ − cosh θ)|2 T 2 + i(sin θ − sinh θ)|3 T 3 |γ .
By measuring in the {|± }-basis there is a probability that we will measure the state | + ++ which means that we have achieved matrix exponentiation by arriving at the state | + ++ N e iθT |γ .

B. Measuring probabillity
In order to investigate the probability of measuring the correct state, we consider the state |φ in Eq. (20). In the {|± }-basis it takes the form We wish to measure a state with a coefficient A+B+C+D, and thus we want to measure the state | + ++ . Note that if we chose our |k states as superpositions, such as |1 = a|001 + b|010 + c|100 , then there is no state in the {|± }-basis with a coefficient A + B + C + D, since the normalization then require the B and C coefficients to be normalized by the superposition coefficients a, b, and c, which means that we get an imbalance between the B and C coefficients and the A and D coefficients. We plot the probabilities of measuring the eight states as a function of θ to see how they behave. The result is seen in Fig. 7. Unfortunately we observe that the probability of measuring the state | + ++ decreases exponentially with θ. This supports our previous understanding that we should indeed keep θ small.

VI. CONCLUSION
We have proposed a simple implementation of a controlled iswap -gate, and shown that these exhibit a high fidelity. We have discussed an implementation of our gates using superconducting circuits, however, the implementation is not limited to this scheme. While the difficulty of implementing our gates does increase with the number of controls, we do believe that our gates will be superior in certain types of quantum computations, especially compared to equivalent circuits built from oneand two-qubit gates, which often become quite deep. Our controlled-iswap can easily be extended to swapping between more qubits, such that it is possible to control swapping between three, four and so on qubits. We also propose a quantum circuit for probabilistic exponentiating of non-Hermitian quantum gates, which is exact for cyclic gates and approximately exact given small parameters for all other non-Hermitian gates. These results could enhance the performance of near-term quantum computing experiments on algorithms that require multi-qubit swapping gates and exponentiating of gates.
Following the procedure of Refs. [59,60] we obtain the following Lagrangian from the circuit diagram in Fig. 3 where the first and third summation is understood as the summation over T 1, T 2, 1, 2 . . . , n. The first line of terms comes from the capacitors and are interpreted as kinetic terms, while the remaining terms come from the Josephson junctions and are interpreted as potential terms. The n indicates the number of blue islands on the circuit diagram, i.e., for the ciswap in Fig. 3(b) n = 1. The capacitance matrix in this case is For the c 2 iswap (see Fig. 8(b) for circuit diagram of this gate) we need n = 2 and the capacitance matrix takes the form K =    C 1 + 2C z,1 0 −C z,1 −C z,1 0 C 1 + 2C z,2 −C z,2 −C z,2 −C z,1 −C z,2 C T 1 + C z,1 + C z,2 + C x −C x −C z,1 −C z,2 −C x C T 2 + C z,1 + C z,2 + C x    , and so on for higher n. The typical transmon have a charging energy much smaller than the junction energy, and therefore the phase is well localized near the bottom of the potential. We can therefore expand the potential part of the Hamiltonian to fourth order U (ϕ) = n i=T 1,T 2,1 By collecting terms we can write the full Hamiltonian as H = n i=T 1,T 2,1