## Abstract

To do large-scale quantum information processing, it is necessary to control the interactions between individual qubits while retaining quantum coherence. To this end, superconducting circuits allow for a high degree of flexibility. We report on the time-domain tunable coupling of optimally biased superconducting flux qubits. By modulating the nonlinear inductance of an additional coupling element, we parametrically induced a two-qubit transition that was otherwise forbidden. We observed an on/off coupling ratio of 19 and were able to demonstrate a simple quantum protocol.

Macroscopic quantum coherence of superconducting structures is an intriguing physical phenomenon. The reasons for studying it include not only the prospect of constructing a quantum computer but also the possibility of realizing quantum-mechanical coherence in artificially fabricated structures. The properties of superconducting qubits (*1*–*3*) are being vigorously studied, with the relevant coherence times of the quantum states now extending to the microsecond range (*4*–*7*), in particular because the qubits are being operated at optimal bias points (*4*). When two or more qubits are coupled, quantum mechanics predicts that the combination can be, roughly speaking, more than the sum of its constituents. This entirely nonclassical concept is called entanglement. The existing quantum algorithms rely heavily on the use of entangled states. With tunable couplings between individual qubits, the design of control pulses for even a large set of qubits is relatively straightforward because the system can be divided into small noninteracting blocks. With nontunable couplings, as in nuclear magnetic resonance quantum computing, a large fraction of the quantum operations performed has to be devoted to effective decoupling procedures (*8*). Recently, several experiments on coupled superconducting qubits have been carried out with both fixed (*9*–*15*) and tunable (*16*–*18*) coupling. Unfortunately, most coupling schemes for superconducting qubits are efficient only away from the optimal point, which results in a shorter coherence time. Thus, realizing tunable coupling at the coherence optimal bias point is crucial for future scalability.

We studied superconducting flux qubits (*19*), which are superconducting loops interrupted by Josephson junctions. When a magnetic flux Φ_{j} close to half-flux quantum Φ_{0}/2 is applied through the loop of qubit *j*, the two lowest-energy states have the supercurrents ±*I*_{pj} rotating in opposite directions. The magnetic energy difference ϵ_{j} = 2*I*_{pj}(Φ_{j} – Φ_{0}/2) between these states can be controlled via Φ_{j}. The tunnelling energy between the states is Δ_{j}. The qubit states have eigenenergies ±ω_{j}/2, where , as shown schematically in Fig. 1A. Resonant control of qubits is possible by modulating ϵ_{j} at the frequency ω_{j}. At the flux optimal point Φ_{j} = Φ_{0}/2, the eigenstates are equal superpositions of the circulating current states, and therefore the expectation value of current vanishes. At this point, the quantum coherence is found to be far superior (*6*, *7*) to the case where Φ_{j} ≠ Φ_{0}/2, because *d*ω_{j}/*d*Φ_{j} = 0 and thus the accumulated quantum phase proportional to the time integral of ω_{j} is insensitive to flux noise.

The natural coupling between qubits *k* and *l* is via mutual inductance *M*_{kl} > 0, resulting in the antiferromagnetic coupling *J*_{kl} = *M*_{kl}*I*_{pk}*I*_{pl}. A collection of *n* qubits can be described by the Hamiltonian (1) where and are Pauli matrices operating on qubit *j*. Owing to vanishing persistent currents (or the off-diagonal coupling term), the coupling only has a weak second-order effect at the optimal point (ϵ_{k} = ϵ_{l} = 0) if *J*_{kl} ≪ |Δ_{k}–Δ_{l}|. It is thus easy to decouple qubits at the optimal point. To realize universal two-qubit gates (that is, to turn the coupling on), it would be ideal to drive either the |00 〉↔|11 〉 or the |10 〉↔|01 〉 transition, where |*jk* 〉 are the two-qubit eigenstates. This results in a gate equally as efficient as the controlled-NOT (CNOT). However, simply driving the fluxes Φ_{1} and Φ_{2} with microwaves in resonance with such transitions cannot realize the desired operation because the microwave essentially couples through single-qubit operators . In general, the corresponding transitions are forbidden for any value of *J*_{kl}, although the eigenstates are modified. Different ways to overcome this problem are being pursued (*20*–*23*).

Our approach (*22*) is to couple two qubits via a third adiabatic qubit (*n* = 3) with higher Δ_{3}, which results in an effective coupling between qubits 1 and 2 given by (2)

The coupling can be interpreted as a sum of direct inductive coupling *J*_{12} and indirect coupling via the nonlinear ground-state inductance *L*_{Q} = –2(*d*^{2}ω_{3}/*d*Φ_{3}^{2})^{–1} of qubit 3. Applying a microwave through the loop of qubit 3 at the frequency (Δ_{2} ∓ Δ_{1})/*h* modulates *L*_{Q} via ω_{3} and results in a term proportional to σ_{x}^{1}σ_{x}^{2} ± σ_{y}^{1}σ_{y}^{2} in the rotating frame for which the desired transitions are allowed. The induced coupling can be thus attributed to the parametric modulation of the coupling energy at a microwave frequency. In the linear approximation, the two-qubit oscillations have the frequency Ω_{12}/*h* = (*dJ*_{12}^{eff}/*d*ϵ_{3})δϵ_{3}/*h*, where δϵ_{3} is the amplitude of the microwave driving applied to qubit 3. This has a maximum around ϵ_{3} = ±Δ_{3}/2 and vanishes at ϵ_{3} = 0 and at |ϵ_{3}| ≫ Δ_{3}. We focus here on the sum-frequency transition |00 〉↔|11 〉. The functioning of the tunable parametric coupling is very convenient. When the microwave at the sum frequency is off, the coupling is also off. Vice versa, when the microwave is turned on, the coupling goes on. The reduced two-qubit rotating-frame Hamiltonian (3) where Ω_{j} is the (resonant) microwave-induced single-qubit Rabi frequency of qubit *j* and ϕ_{j} is the microwave phase, offers full control and can in principle be set to zero by turning off all microwaves.

Figure 1 describes our experimental setup. By using the kinetic inductance of shared superconducting wires, we couple two qubits (Δ_{1} < Δ_{2}) to a third qubit with a larger gap Δ_{3}. Direct coupling *J*_{12} is expected to be small. On the basis of a series of spectroscopic measurements (*24*), we know that we can rely on the accuracy of lithography and have very similar areas (within ∼0.5% difference) of qubits 1 and 2. Therefore, we have Φ_{1} ≈ Φ_{2} even with a single uniform flux bias, whereas Φ_{3} is deliberately offset to have a finite Ω_{12}. We can write Φ_{j} = Φ_{3} + ΔΦ_{j}, where ΔΦ_{j} (*j* = 1,2) is a small offset. To manipulate the qubits, we apply microwaves of controllable duration via an on-chip microwave line. During the manipulation with the microwaves that are assumed to couple to all loops, we apply a small bias current (a few percent of critical current) through the readout superconducting quantum interference device (SQUID) to fine-tune the biases of both qubits 1 and 2 to be at the optimal point. Figure 2 illustrates the measured spectrum when the minima of the qubit transition frequencies are aligned. Qubits 1 and 2 have sufficiently different gaps as designed, and a sign of the desired sum-frequency transition is also visible.

To show that the coupling scheme is indeed tunable and coherent, we have to be able to set the rotating-frame Hamiltonian to zero as well as to drive the single-qubit transitions in time domain regardless of the input state (off). Also, we have to be able to drive, in a time domain, the two-qubit transition (on). Figure 3 shows measured examples of this at the combined optimal point of qubits 1 and 2. It is clear that a single-qubit transition frequency does not depend significantly on the state of the other. On the other hand, we can drive the sum-frequency transition as fast as Ω_{12}^{max}/*h* = 23.2 MHz, which is obtained with the maximum power we can apply with the present microwave source and attenuation of our setup. The minimum time for a universal gate is given by half of the period; that is, 22 ns. A quantitative measure of the on/off ratio is obtained by comparing the maximal two-qubit oscillation frequency to the frequency at which unwanted coupling takes place. Using a Ramsey fringe measurement of the different transitions, we find that the Hamiltonian in the energy eigenbasis (and thus the logical rotating-wave Hamiltonian) has a spurious term –(κ/2)σ_{z}^{1}σ_{z}^{2}, with κ/*h* = 1.23 MHz, which is quite small compared to the single-qubit frequencies of about 4.000 GHz and 6.889 GHz. Because this term results in unwanted phase rotation at the frequency κ/*h*, we conclude that the demonstrated on/off ratio of our coupling scheme is Ω_{12}^{max}/κ ≈ 19. In addition to the high degree of tunability, the coherence properties are quite good, as characterized by the relaxation time *T*_{1} and the Ramsey dephasing time *T*_{2}^{Ramsey}. Qubit 2 has about *T*_{1} = 1.0 μs and *T*_{2}^{Ramsey} = 0.8 μs, whereas qubit 1 has *T*_{1} = 0.3 μs and *T*_{2}^{Ramsey} = 0.2 μs. The lifetime of the state |11 〉 is limited by qubit 1, and we get *T*_{2}^{Ramsey} = 0.2 μs for the sum-frequency transition. The reduced coherence time of qubit 1 is caused by a resonance close by. The fact that the resonant frequency of the two-qubit transition is reduced at large driving amplitudes can be understood from simulation as the effect of unwanted coupling of the microwave to qubits 1 and 2(*6*, *25*), which is unavoidable in the present sample design.

As further proof of the functioning of the scheme and the good coherence properties combined with a high on/off ratio, we have demonstrated a simple multipulse quantum protocol related to quantum coin tossing (*26*). One application of quantum coin tossing is to store classical information in a way that is immune to certain types of intervention and noise. For example, if one stores classical bits on eigenstates of σ_{x}, then these bits are immune to the action of a malicious quantum hacker who breaks into one's quantum computer and flips bits about the *x* axis: Because of the coding, the hacker may think he or she is flipping bits but is in fact only applying a global phase to each bit. Our protocol is complementary to this: We detect the presence of a classically benign hacker, who performs an operation whose sole effect is to multiply certain logical states by a phase. Such a phase is classically undetectable: The operation of the benign hacker on classical bits is to apply a global phase. Quantum-mechanically, however, one can detect this hacker by using superposition and entanglement. Figure 4 shows the performance of the protocol.

Driving at the sum-frequency transition induces the unitary gate *U*_{sum} = exp[–*i*θ/4(σ_{x}^{1}σ_{x}^{2}–σ_{y}^{1}σ_{y}^{2})], where θ is varied by the microwave duration. When θ = π, the gate is equivalent to the double-CNOT (or the iSWAP) up to single-qubit rotations. Three applications of this gate suffice to generate any two-qubit gate when supplemented by single-qubit gates (*27*). At θ = 2π, the gate is diagonal with the entries (–1,1,1,–1), and if the input state is an eigenstate, the resulting oscillation thus has a 2π period. But the underlying period is actually 4π, which can be revealed using superposition states. We do this by applying a π/2 rotation on qubit 2 before and after the application of *U*_{sum}. The experimental confirmation of this purely quantum-mechanical effect is shown in Fig. 4, where we see that the oscillation frequency is halved and phase-shifted when π/2 rotations are used. There is no analogy for this effect for a single qubit, because there the phase is just global and undetectable. The measurement result demonstrates the familiar quantum fact that you have to rotate by 4π, not 2π, to get back to where you started and is clear evidence of the entanglement inherent in the correlations between the qubits during their time evolution.

Our experiments demonstrate that the tunable coupling between flux qubits can be realized entirely using the application of microwaves at the coherence optimal point, which is very important for scalability. This indicates that parametric couplers are strong candidates as fundamental building blocks of gate-based quantum computers, but optimization of the scheme requires further study. We expect the coherence time of tunably coupled qubits to reach the same level as that of individual optimized qubits when, for example, independent readouts allowing for bias optimization (*6*, *7*) are combined with this coupling scheme in a future device.