Fast Multiqubit Gates through Simultaneous Two-Qubit Gates

Near-term quantum computers are limited by the decoherence of qubits to only being able to run low-depth quantum circuits with acceptable fidelity. This severely restricts what quantum algorithms can be compiled and implemented on such devices. One way to overcome these limitations is to expand the available gate set from single- and two-qubit gates to multiqubit gates, which entangle three or more qubits in a single step. Here, we show that such multiqubit gates can be realized by the simultaneous application of multiple two-qubit gates to a group of qubits where at least one qubit is involved in two or more of the two-qubit gates. Multiqubit gates implemented in this way are as fast as, or sometimes even faster than, the constituent two-qubit gates. Furthermore, these multiqubit gates do not require any modification of the quantum processor, but are ready to be used in current quantum-computing platforms. We demonstrate this idea for two specific cases: simultaneous controlled-$Z$ gates and simultaneous iswap gates. We show how the resulting multiqubit gates relate to other well-known multiqubit gates and demonstrate through numerical simulations that they would work well in available quantum hardware, reaching gate fidelities well above 99%. We also present schemes for using these simultaneous two-qubit gates to swiftly create large entangled states like Dicke and Greenberger-Horne-Zeilinger states.

Popular Summary

Quantum algorithms may outperform classical ones on important computational tasks in chemistry, optimization, and many other fields. However, to run on current quantum computers, these algorithms must be compiled into long sequences of elementary operations (gates) on one or two qubits. Since the available quantum hardware still struggles to protect qubits from noise, it is desirable to execute the algorithms swiftly to have a hope for quantum advantage. Here we show how two-qubit gates on existing quantum hardware can be run simultaneously to create new multi-qubit gates, which are both fast and powerful: they entangle more qubits and take less time to execute than the two-qubit gates.

The key to creating three-qubit gates in our scheme is two-qubit gates that rely on swapping excitations between neighboring qubits. When the middle qubit in a chain of three qubits interacts in that manner with both its neighbors, a pathway is created for swapping excitations between the two outer qubits, conditioned on the state of the middle qubit. We demonstrate the construction of such three-qubit gates for different types of two-qubit gates and for multiple architectures, showing through extensive numerical simulations that they can be implemented with high fidelity in available experimental setups.

Our results suggest that additional multi-qubit gates can be discovered using similar constructions with other two-qubit gates. Having such new multi-qubit gates on current quantum computers opens up for re-compiling many quantum algorithms into shorter gate sequences, enhancing the performance of the quantum computers without needing to upgrade the hardware.

Figure 1

Setup and operation for the three-qubit gate realized through simultaneous application of two cz gates. (a) The setup considered is a linear chain of three qubits with nearest-neighbor coupling. Going from left to right in the chain, we denote the qubits $q_1$, $q_0$, and $q_2$. The cz gates ${\mathrm{CZ}}_{0j}$ between qubits $0$ and $j=\{1,2\}$ are applied simultaneously by activating a coupling between the $|1_0{1}_j⟩$ and $|2_0{0}_j⟩$ states with the coupling strength ${\lambda }_j(t)$. (b) The transitions in the three-qubit system activated by the application of the cz gates. With the three-qubit states denoted by $|q_0{q}_1{q}_2⟩$, the transitions $|11x⟩\leftrightarrow |20x⟩$ with $x=\{0,1\}$ are activated by ${\mathrm{CZ}}_{01}$ (red), and the transitions $|1x1⟩\leftrightarrow |2x0⟩$ are activated by ${\mathrm{CZ}}_{02}$ (purple). We assume that both cz gate operations are detuned by $\delta$ from resonance. (c) We denote the three-qubit operation resulting from the simultaneous application of the two cz gates by cczs (controlled-czs), since it applies both cz and swap gates to the target qubits $q_1$ and $q_2$ conditioned on the control qubit $q_0$.

Figure 2

Decomposition of the three-qubit cczs gate into two-qubit gates. Note that, for the case of a linear chain [see Fig. 1], the control qubit $q_0$ in the cczs gate is the middle qubit, but the decomposition into two-qubit gates requires $q_1$ to be the middle qubit, since it has to interact with both $q_0$ and $q_2$.

Figure 3

Converting cczs gates into other three-qubit gates. (a) Constructing a Fredkin gate with a $\mathrm{cczs}(\pi /2,0,0)$ gate and a ccz gate. (b) Constructing an iFredkin gate from a $\mathrm{cczs}(\pi /2,\pi /2,0)$ gate and a cz gate.

Figure 4

Quantum circuits for generating the three-qubit GHZ state $(|000⟩+|111⟩)/\sqrt{2}$. (a) A circuit using single-qubit gates and two-qubit cz gates. (b) A circuit using single-qubit gates and one cczs gate.

Figure 5

The steps for preparing the state $(\sqrt{3/5})|1_0⟩|D_4^2⟩+(\sqrt{2/5})|0_0⟩|D_4^1⟩$ using the generalization of the cczs gate to a five-qubit system.

Figure 6

Creating large $\mathrm{W}$ states rapidly on a square grid of qubits. We first prepare qubit $0$ ($q_0$) in its second excited state and carry out the rest of step 1 in Sec. 2e2 to create $|D_4^1⟩$ on the neighboring qubits (red circles). Next, we swap each of the neighboring qubits with the qubit next to them that is farthest from the center qubit (blue circles except $q_0$). Flipping the $|1⟩$ part of the state for these new center qubits to $|2⟩$ and having them interact with their nearest neighbors creates $|D_4^1⟩$ on those neighbors (purple and red circles). For brevity, the new cells starting from $q_3$ and $q_4$ are not shown. The result after these steps (two rounds of single or simultaneous single-qubit operations, three rounds of simultaneous two-qubit gates) is the 16-qubit $\mathrm{W}$ state $|D_{16}^1⟩$.

Figure 7

Implementing a cczs gate with tunable qubits. (a) A sketch of the setup, with qubit numbering following the convention established in Fig. 1. The superconducting qubits are transmon qubits [103], which are nonlinear $LC$ oscillators where the nonlinear inductances are provided by Josephson junctions (boxes with crosses in the sketch). When two Josephson junctions are combined in a loop [a superconducting quantum interference device (SQUID)], the effective inductance, and thus the qubit frequency, can be tuned by controlling the magnetic flux through the loop. (b) The tuning of the qubit energies used to implement a high-fidelity $\mathrm{cczs}(\pi /2,\pi ,0)$ gate. (c) Population of the states $|110⟩$, $|200⟩$, and $|101⟩$ during the $\mathrm{cczs}(\pi /2,\pi ,0)$ gate when the initial state is $|110⟩$. As Eq. (24) shows, the main effect of this gate is to swap the population between qubits 1 and 2 if qubit 0 is in its excited state.

Figure 8

Decomposition of the $\mathrm{cczs}(\pi /2,\pi ,0)$ gate into single-qubit and cz gates, obtained using Qiskit [154]. For the single-qubit gates, we use the notation $S=\sqrt{Z}=Z_{\pi /2}$ and $\sqrt{X}=X_{\pi /2}$. Note that this decomposition requires qubit 2 being placed in the middle of the linear chain, since it has to perform cz gates with both qubit 0 and qubit 1.

Figure 9

Setup for implementing the cczs gate with tunable couplers $c_1$ and $c_2$ connecting the qubits. The frequency of each coupler $j$ is tuned by changing the magnetic flux ${\mathrm{\Phi }}_j$ through the loop formed by the two Josephson junctions.

Figure 10

Calibrating the cczs gate with tunable couplers. (a) Population in $|200⟩$ as a function of gate time and the detuning between the modulation frequency ${\omega }_{{\mathrm{\Phi }}_1}$ and the frequency of the transition $|110⟩\leftrightarrow |200⟩$ when calibrating the ${\mathrm{CZ}}_{01}$ gate by initializing the system in $|110⟩$. (b) The gate fidelity for the ${\mathrm{CZ}}_{01}$ gate for the parameters in (a). (c) Population in $|200⟩$ as a function of gate time and the detuning between the modulation frequency ${\omega }_{{\mathrm{\Phi }}_2}$ and the frequency of the transition $|101⟩\leftrightarrow |200⟩$ when calibrating the ${\mathrm{CZ}}_{02}$ gate by initializing the system in $|101⟩$. (d) The gate fidelity for the ${\mathrm{CZ}}_{02}$ gate for the parameters in (c). (e) Population in $|110⟩$ as a function of gate time and the detuning between the modulation frequency ${\omega }_{{\mathrm{\Phi }}_1}$ and the frequency of the transition $|110⟩\leftrightarrow |200⟩$ when calibrating the $\mathrm{cczs}(\pi /2,\pi ,0)$ gate by initializing the system in $|101⟩$. During the calibration, we also vary ${\omega }_{{\mathrm{\Phi }}_2}$, but in this plot it is kept fixed. (f) The gate fidelity for the $\mathrm{cczs}(\pi /2,\pi ,0)$ gate for the parameters in (e).

Figure 11

Calibrating the $\mathrm{cczs}(\pi /2,\varphi ,0)$ gate by tuning the phase difference ${\phi }_1-{\phi }_2$ between the signals modulating the two tunable couplers. The simulations are performed using three energy levels per qubit. The plots shows the gate fidelity for the $\mathrm{cczs}(\pi /2,\varphi ,0)$ gate as a function of $\varphi$ and ${\phi }_1-{\phi }_2$. All other parameters are the same as those that gave the highest gate fidelity for $\mathrm{cczs}(\pi /2,\pi ,0)$ in Fig. 10. For each value of $\varphi$ shown in the plot, the highest gate fidelity exceeds 99%.

Figure 12

Setup and operation for the three-qubit gate realized through simultaneous application of two iswap gates. (a) The setup considered is a linear chain of three qubits with nearest-neighbor coupling. The iswap gates $\mathrm{i}{\mathrm{swap}}_{0j}$ between qubits $0$ and $j=\{1,2\}$ are applied simultaneously by activating couplings between the $|1_0{0}_j⟩$ and $|0_0{1}_j⟩$ states with coupling strength $g_j$. (b) The transitions in the three-qubit system activated by the application of the iswap gates. The states $|000⟩$ and $|111⟩$ are not affected by the gates. (c) We denote the three-qubit operation resulting from the simultaneous application of the two iswap gates by div, since it is a “divider” gate that can distribute one or two excitations among all three qubits.

Figure 13

Quantum circuit for generating the three-qubit $W$ state $(|100⟩+|010⟩+|001⟩)/\sqrt{3}$ using the div gate. The phase gate $S$ is $\sqrt{Z}$.

Figure 14

Implementing the $\mathrm{div}(\pi /4,\pi /2)$ gate with tunable qubits. (a) The tuning of the qubit energies used to realize the gate. (b) Population of the states $|100⟩$, $|010⟩$, and $|001⟩$ (the single-excitation subspace) as a function of time during the gate when the initial state is $|010⟩$. As Eq. (61) shows, the effect of the gate is to divide an excitation in one of the outer qubits such that half ends up in the middle qubit and a quarter each in the outer qubits.

Figure 15

Population transfers for different initial states [(a) $|100⟩$, (b) $|010⟩$, (c) $|001⟩$] in the one-excitation subspace in the implementation of the cczs gate with tunable qubits. All simulation parameters and notation conventions are the same as for Fig. 7 in the main text.

Figure 16

Population transfers for different initial states [(a) $|110⟩$, (b) $|101⟩$, (c) $|011⟩$] in the two-excitation subspace in the implementation of the div gate with tunable qubits. All simulation parameters and notation conventions are the same as for Fig. 14 in the main text.