Flip-chip architectures have recently enabled significant scaling-up of multi-qubit circuits and have been used to assemble hybrid quantum systems that combine different substrates, for example, for quantum acoustics experiments. The standard flip-chip method uses superconducting galvanic connections between two substrates, typically implemented using sophisticated indium wafer-bonding systems, which give highly reliable and temperature-cyclable assemblies, but are expensive, somewhat inflexible in design, and require robust substrates that can sustain the large compressive forces required to cold-weld the indium bonds. A much simpler method is to assemble dies using very low-force contacts and air-dried adhesives, although this does not provide a galvanic contact between the dies. Here, we demonstrate that the latter technique can be used to reliably couple superconducting qubit circuits, in which the qubits are on separate dies, without the need for a galvanic connection. We demonstrate full vector qubit control of each qubit on each of the two dies, with high-fidelity single-shot readout, and further demonstrate entanglement-generating excitation swaps as well as benchmark a controlled-Z entangling gate between the two qubits on the two dies. This exemplifies a simple and inexpensive assembly method for two-plus-one-dimensional quantum circuit integration that supports the use of delicate or unusually shaped substrates.

Superconducting qubits are a promising platform for the realization of quantum computing technologies.^{1–3} As the size and complexity of superconducting quantum processors increases, the need for more flexible architectures and increased connectivity has grown.^{4–7} Achieving sufficient connectivity can be achieved using through-chip vias^{8,9} or flip-chip assemblies with cold-welded indium bump bonding.^{10} These sophisticated methods require expensive tooling inaccessible to most academic researchers. To address these challenges, we have previously developed and described a facile flip-chip fabrication process,^{11} originally conceived and developed for coupling superconducting qubit circuits to quantum acoustic systems and other hybrid applications. Here, we expand on this technique and demonstrate that it can also serve for the high-fidelity coupling of superconducting qubit circuits, each circuit fabricated on a separate die, without the need for a galvanic connection between the dies. This holds promise for the simple and rapid expansion of qubit circuit complexity, allowing wiring in two-plus-one dimensions and, thereby, alleviating qubit interconnect challenges.

The device measured in this work is shown in Fig. 1. The device comprises two double-side polished sapphire dies, where at least one die must be transparent, a larger control die with qubit *Q*1 shown prior to assembly in panel (a), and a second smaller die with qubit *Q*2 shown in panel (c), with the assembled dies shown in panel (b), with the smaller die outlined in red. The dies are assembled using a mask aligner, achieving horizontal alignment better than 2 *μ*m using alignment marks (see Fig. 1). Vertical spacing is set by $7\u2009\mu $m high SU-8 photodefined spacers, and dies are attached using photoresist as an adhesive, which is briefly cured with a hot-air gun; spacer height is verified by profilometry and assembled spacing by electron microscopy. Qubits and control wiring are patterned in aluminum, with Josephson junctions fabricated using the Dolan technique,^{12} using qubits with two small-area Josephson junctions in a superconducting quantum interference device (SQUID) loop, allowing frequency tuning using a flux-bias Z control line.^{13,14} Excitation pulses are through a separate XY control line and readout uses a quarter-wave coplanar resonator dispersively coupled to the qubit, probed by a readout transmission line.^{15} Control and readout lines for the qubit on the smaller die are located on the larger control die. All the galvanic control and readout connections are made using Al wire bonds to bond pads on the periphery of the control die. Signals are coupled to the flipped die using vacuum-gapped fixed inductive and capacitive couplers, shown in the equivalent circuit in panel (e). Using this assembly, we demonstrate quantum control and single-shot readout of each qubit, high-fidelity two-qubit entanglement, as well as a controlled-Z gate between the two qubits, which we characterize using quantum process tomography and randomized benchmarking.

The qubits on the two dies are coupled to one another through direct vacuum–gapped inductive coupling, achieved by aligning sections of coplanar transmission line with a $10\u2009\mu $m wide center trace and a $50\u2009\mu $m gap on each die (see Fig. 1). Each transmission line is shorted to ground on one end and connected to the SQUID loop of its respective qubit at the other end. The coupling inductors overlap when the flip-chip is assembled, a method that gives large and reliable mutual coupling between the elements on each die.^{16,17}

Unique to the device here is the control line geometry for *Q*2, with no galvanic ground to the flipped die. This qubit, which also includes a two-junction SQUID loop, is tuned using a Z-control transmission line shorted to ground, located on the control die directly beneath *Q*2's SQUID loop, inductively coupled with a design inductance of $\u22483$ pH. The qubit is excited using on-resonant pulses coupled via a second XY-control transmission line terminated in an open circuit, located on the larger die beneath the capacitive cross of *Q*2, capacitively coupled with a design capacitance of $\u224813$ aF. The shape and location of these control lines, which sets their coupling strength, are determined using finite element simulations (Sonnet Software, Syracuse, NY). We use unique inductive qubit–qubit coupling and inductive coupling between the bottom die readout line and the flipped die *Q*2 readout resonator; see Table I. Simulations suggest that small flip-chip alignment errors do not contribute significantly.^{11} For the qubit–qubit coupling of interest here, simulations indicate that a misalignment by $2\u2009\mu $m in the lateral directions alters the coupling by at most 2%, while a height offset of $0.5\u2009\mu $m alters the coupling by at most 3%.

Parameter . | Qubit 1 . | Qubit 2 . |
---|---|---|

Designed qubit capacitance (fF) | 88 | 88 |

SQUID inductance (nH) | 12.1 | 13.5 |

Qubit maximum frequency (GHz) | 4.65 | 4.39 |

Qubit anharmonicity, α (MHz) | −179 | −190 |

Qubit lifetime, T_{1}, at maximum frequency point (μs) | 7.7 ± 0.1 | 17.3 ± 0.1 |

Qubit lifetime, T_{1}, at operating point (μs) | 16.4 ± 0.1 | |

Qubit Ramsey dephasing time, T_{2}, at max frequency (μs) | 10.1 ± 0.2 | 7.5 ± 0.1 |

Qubit Ramsey dephasing time, T_{2}, at operating point (μs) | 1.6 ± 0.1 | |

Readout resonator frequency (GHz) | 5.46 | 5.63 |

Designed qubit–readout resonator coupling (MHz) | 35 | 35 |

Designed readout dispersive shift (MHz) | 0.14 | 0.12 |

Measured readout dispersive shift (MHz) | 0.2 | 0.1 |

Designed qubit–qubit coupling (MHz) | 13 | |

Measured qubit–qubit coupling (MHz) | 12 |

Parameter . | Qubit 1 . | Qubit 2 . |
---|---|---|

Designed qubit capacitance (fF) | 88 | 88 |

SQUID inductance (nH) | 12.1 | 13.5 |

Qubit maximum frequency (GHz) | 4.65 | 4.39 |

Qubit anharmonicity, α (MHz) | −179 | −190 |

Qubit lifetime, T_{1}, at maximum frequency point (μs) | 7.7 ± 0.1 | 17.3 ± 0.1 |

Qubit lifetime, T_{1}, at operating point (μs) | 16.4 ± 0.1 | |

Qubit Ramsey dephasing time, T_{2}, at max frequency (μs) | 10.1 ± 0.2 | 7.5 ± 0.1 |

Qubit Ramsey dephasing time, T_{2}, at operating point (μs) | 1.6 ± 0.1 | |

Readout resonator frequency (GHz) | 5.46 | 5.63 |

Designed qubit–readout resonator coupling (MHz) | 35 | 35 |

Designed readout dispersive shift (MHz) | 0.14 | 0.12 |

Measured readout dispersive shift (MHz) | 0.2 | 0.1 |

Designed qubit–qubit coupling (MHz) | 13 | |

Measured qubit–qubit coupling (MHz) | 12 |

Single qubit gate . | Duration . | Qubit 1 fidelity . | Qubit 2 fidelity . |
---|---|---|---|

I | 50 ns | 0.998 0 ± 0.000 3 | 0.997 1 ± 0.000 4 |

X/2 | 25 ns | 0.997 7 ± 0.000 1 | 0.997 9 ± 0.000 3 |

−X/2 | 25 ns | 0.997 4 ± 0.000 1 | 0.998 3 ± 0.000 5 |

Y/2 | 25 ns | 0.998 3 ± 0.000 1 | 0.997 7 ± 0.000 4 |

−Y/2 | 25 ns | 0.998 1 ± 0.000 1 | 0.997 1 ± 0.000 5 |

Z/2 | 25 ns | 0.998 3 ± 0.000 1 | 0.999 5 ± 0.000 3 |

X | 50 ns | 0.996 7 ± 0.000 1 | 0.993 2 ± 0.000 5 |

−X | 50 ns | 0.996 5 ± 0.000 2 | 0.996 4 ± 0.000 3 |

Y | 50 ns | 0.996 6 ± 0.000 1 | 0.994 8 ± 0.000 5 |

−Y | 50 ns | 0.997 6 ± 0.000 2 | 0.997 1 ± 0.000 4 |

Z | 25 ns | 0.998 4 ± 0.000 1 | 0.999 3 ± 0.000 4 |

H | 50 ns | 0.996 3 ± 0.000 2 | 0.996 7 ± 0.000 5 |

Single qubit gate . | Duration . | Qubit 1 fidelity . | Qubit 2 fidelity . |
---|---|---|---|

I | 50 ns | 0.998 0 ± 0.000 3 | 0.997 1 ± 0.000 4 |

X/2 | 25 ns | 0.997 7 ± 0.000 1 | 0.997 9 ± 0.000 3 |

−X/2 | 25 ns | 0.997 4 ± 0.000 1 | 0.998 3 ± 0.000 5 |

Y/2 | 25 ns | 0.998 3 ± 0.000 1 | 0.997 7 ± 0.000 4 |

−Y/2 | 25 ns | 0.998 1 ± 0.000 1 | 0.997 1 ± 0.000 5 |

Z/2 | 25 ns | 0.998 3 ± 0.000 1 | 0.999 5 ± 0.000 3 |

X | 50 ns | 0.996 7 ± 0.000 1 | 0.993 2 ± 0.000 5 |

−X | 50 ns | 0.996 5 ± 0.000 2 | 0.996 4 ± 0.000 3 |

Y | 50 ns | 0.996 6 ± 0.000 1 | 0.994 8 ± 0.000 5 |

−Y | 50 ns | 0.997 6 ± 0.000 2 | 0.997 1 ± 0.000 4 |

Z | 25 ns | 0.998 4 ± 0.000 1 | 0.999 3 ± 0.000 4 |

H | 50 ns | 0.996 3 ± 0.000 2 | 0.996 7 ± 0.000 5 |

Two qubit gate . | Duration . | . | Fidelity . |
---|---|---|---|

Controlled Z | 40 ns | 0.940 ± 0.007 |

Two qubit gate . | Duration . | . | Fidelity . |
---|---|---|---|

Controlled Z | 40 ns | 0.940 ± 0.007 |

The device was operated on the mixing plate of a dilution refrigerator with a base temperature of 7 mK. Control and readout uses custom electronics coupled through heavily filtered and attenuated (for control signals) coaxial cables. Readout signals are amplified by a traveling wave parametric amplifier^{18} (TWPA) followed by a 4 K cryogenic amplifier; see Ref. 15 for more details. In Fig. 2(a), we show the microwave transmission through the readout line, with the two readout resonators visible as dips in transmission at 5.5 and 5.6 GHz. The dip at 6.4 GHz is due to the TWPA. Inflections at 3.5 and 7.5 GHz are due to filters and amplifier bandwidths. We do not observe any extra box- or slot-mode resonances due to the flip-chip geometry. Modeling the geometry as a rectangular microstrip antenna, we calculate the lowest frequency resonant mode for the 2 × 4 mm^{2} flipped die to be at $\u224837$ GHz, well above the frequencies used here. Larger flipped dies reduce the resonant frequency, for example, for a 20 × 20 mm^{2} die; the lowest resonance would be at $\u22487.5$ GHz.^{19}

We can excite the qubits using a resonant microwave pulse applied to their XY control lines on the control die. Readout of the qubit state is performed by probing the dispersively coupled readout resonator.^{20–22} This coupled resonator shows a shift in its resonance frequency when the qubit is excited, shown for *Q*2 in Fig. 2(b). After exciting the qubits, we can monitor their decay to the ground state, yielding the qubit energy lifetime *T*_{1}, shown in Fig. 2(c), where fits yield $T1=8$ and $16\u2009\mu $s for *Q*1 and *Q*2, respectively. We observe a residual thermal population in *Q*2 of about 10%, possibly due to poor thermalization, as this chip is thermally connected only by photoresist adhesive.

We perform a Ramsey experiment with a small detuning, shown in the lower panel of Fig. 2(c), to measure the qubit's energy coherence time *T*_{2}. Fitting yields $T2=10$ and $2\u2009\mu $s for *Q*1 and *Q*2, respectively, measured at the operating point for the CZ gate, with *Q*1 at its maximum frequency point and *Q*2 flux-biased away from its maximum frequency. An independent measurement of *Q*2 performed at its maximum frequency point yields $T2=7\u2009\mu $s. These times are in line with our standard (non-floating) qubits, where we find *T*_{1} times of order $5\u221220\u2009\mu $s and operating point *T*_{2} times of order $0.5\u22122\u2009\mu $s.

We also tune up single qubit rotations for both qubits and benchmark the standard single qubit gates using randomized benchmarking,^{23} as shown in Table I. We find the fidelity of all single qubit rotations to be well above 0.99.

We next investigate controlled interactions between the two qubits. By first exciting one qubit and then bringing the qubits into frequency resonance, we can swap states between them. By varying the amount of time that the two qubits are tuned into resonance, we can observe Rabi swaps, shown in Fig. 3(a). We perform two-qubit state tomography^{24} to extract density matrices for our system and can calculate fidelities from the trace distance to the ideal state density matrices. The density matrices of the two-qubit state before and after a full swap are shown in Figs. 3(b) and 3(c); we find a state fidelity of $0.96\u2009\xb1\u20090.02$ for the two-qubit state before the swap and $0.94\u2009\xb1\u20090.01$ after the swap. By tuning the qubits into resonance for one-half of a full swap, we can generate a Bell state $(|ge\u27e9\u2212|eg\u27e9)/2$ between them. The Bell state density matrix measured using state tomography is shown in Fig. 3(d), with a state fidelity of $0.946\u2009\xb1\u20090.005$ after correcting for readout errors.^{25} These results show that the flip-chip construct preserves good qubit performance and that we can separately control each qubit on the two dies as well as generate high-fidelity entanglement between the qubits.

A key gate for quantum computing is the controlled-Z (CZ) gate, in which a relative phase shift of *π* is acquired by the two-qubit $|ee\u27e9$ state compared to the other states in the computational manifold. We implement this gate via an interaction of the $|ee\u27e9$ state with the non-computational $|fg\u27e9$ state, which can be brought into resonance with the $|ee\u27e9$ state using single-qubit Z-tuning pulses. To determine the parameters for this gate, we first perform qubit spectroscopy on *Q*1 at strong drive powers, revealing the energy eigenstates shown in Fig. 4(a). In this experiment, while applying a strong microwave tone to *Q*1 at a frequency shown on the vertical axis, we tune its frequency, shown on the horizontal axis, while leaving *Q*2's frequency fixed. We then measure the transmission of the readout line at a frequency close to *Q*1's readout resonator frequency, where the transmission increases when *Q*1 is excited to higher states. We calculate the eigenmode frequencies for the system, overlaid as black dashed lines in the figure, and find good agreement with the experimental data.

The critical avoided-level crossing for the CZ gate is between the $|ee\u27e9$ and $|fg\u27e9$ states, circled in Fig. 4(a). The CZ gate is implemented by applying a 30 ns-duration Z tuning pulse to *Q*1 and *Q*2, moving the $|ee\u27e9$ state of the two-qubit system into an avoided-level crossing with $|fg\u27e9$, thus enabling a full swap from $|ee\u27e9$ to $|fg\u27e9$ and back. This induces a phase change $|ee\u27e9\u2192ei\pi |ee\u27e9$, leaving the other two-qubit states $|gg\u27e9,|eg\u27e9$ and $|ge\u27e9$ unchanged.

The relative phase of the target $|ee\u27e9$ state can be measured by performing Ramsey experiments on *Q*2 dependent on the state of *Q*1, as shown in Figs. 4(b)–4(d). In Fig. 4(b), no CZ tuning pulse are applied, and the Ramsey measurement of *Q*2 shows the expected non-interacting result. In (c), where CZ pulses are applied, we see that *Q*2's $|e\u27e9$ state acquires a *π* phase shift if *Q*1 is in its $|e\u27e9$ state; in Fig. 4(d), a set of additional pulses are added to compensate for phase accumulation during the gate.

We can calculate the CZ gate *χ* matrix, using $\epsilon (\rho )=\u2211mnE\u0303m\rho E\u0303n\u2020\chi mn$, where *ρ* is the input density matrix and $\epsilon (\rho )$ is the output density matrix.^{26} Here $E\u0303$ is drawn from the basis of Pauli operators. We perform two-qubit process tomography^{27} on the CZ gate; from the experimentally measured *χ* matrix, we calculate a final state fidelity of $0.947\u2009\xb1\u20090.003$ to the ideal *χ* matrix. We also use two-qubit randomized benchmarking to further characterize the CZ gate, by expanding our single-qubit benchmarking procedure to the two-qubit Clifford group.^{28} By compared to a reference measurement taken without interleaved CZ gates, we calculate a benchmarking fidelity of $0.940\u2009\xb1\u20090.007$.

We have demonstrated the use of a non-galvanically connected flip-chip architecture with qubits on both a galvanically grounded control die and an electrically floating flipped die, showing full high-fidelity control of both qubits and coupling between qubits located on separate dies. We also show excitation swaps and high-fidelity two-qubit entanglement between the qubits, demonstrate a controlled-Z gate, and perform randomized benchmarking of single qubit gates as well as process tomography and two-qubit randomized benchmarking of the controlled-Z gate. We did not observe significant crosstalk effects, but, of course, these may occur in more complex circuits. The architecture described here could prove useful for designing and implementing more complex superconducting qubit experiments, enabling a simplification of complicated wiring than could not easily be achieved using a single-die architecture.

The authors thank P. J. Duda for fabrication assistance. The authors thank MIT Lincoln Lab for providing a travelling-wave parametric amplifier (TWPA). Devices and experiments were supported by the Air Force Office of Scientific Research and the Army Research Laboratory. É.D. was supported by LDRD funds from Argonne National Laboratory; A.N.C. was supported in part by the DOE, Office of Basic Energy Sciences. This work was partially supported by the University of Chicago's MRSEC (NSF Award No. DMR-2011854) and made use of the Pritzker Nanofabrication Facility, which receives support from SHyNE, a node of the National Science Foundation's National Nanotechnology Coordinated Infrastructure (NSF Grant No. NNCI1542205), with additional support from NSF QLCI for HQAN (NSF Award No. 2016136).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.