Experimental set-up
Our experiments are performed on a superconducting quantum processor, named Chuang-tzu 2.0, which comprises 78 qubits arranged in a square lattice configuration with 6 rows and 13 columns. Each pair of nearest-neighbour qubits is interconnected by an adjustable coupler, resulting in a total of 137 couplers within the processor. With all 78 superconducting qubits initialized at their idle points, we prepared the initial state using X gates, and the X-gate pulses are optimized to minimize the leakage to higher energy levels, achieving an average gate fidelity of 99.4%. Then, the RMD pulses are applied on all qubits to engineer the Hamiltonian, for experiments of different parameters. The states of all qubits can be read out simultaneously through the transmission lines coupled to readout resonators. All qubit probabilities are corrected to eliminate the measurement errors.
It is noteworthy that the qubit connectivity in our processor, while maintaining a square lattice structure, differs slightly from processors such as Sycamore53, which feature a square lattice with zig-zag edges. We use a tunable coupler with a capacitively connecting pad architecture45, which facilitates a 1,200-μm spacing between adjacent qubits, thereby ensuring sufficient wiring space. The qubits and couplers are fabricated on the qubit layer chip, whereas the control lines, readout lines and readout cavities are integrated into the wiring layer chip. These two chips are interconnected using flip-chip bonding technology. The readout cavities for the six qubits in each column are multiplexed onto a single readout line and are capacitively coupled to the qubits through interfacial capacitance. The control lines are similarly coupled to the qubits and couplers, enabling excitation and biasing via interfacial capacitance and mutual inductance.
RMD pulse
To implement the Hamiltonian described in the main text, the qubit frequency ω q needs to be rapidly modulated between two values ω q = ω c + δh × h 0 for U + , whereas ω q = ω c − δh × h 0 for U − , where ω c is the common qubit frequency. For the frequency-tunable transmon qubit, the relationship between ω q and the amplitude of Z pulse z is
$${\omega }_{{\rm{q}}}=\sqrt{8{E}_{\mathrm{JJ}}\,{E}_{{\rm{C}}}| \cos (kz+b)| }-{E}_{{\rm{C}}},$$ (6)
where E JJ denotes the Josephson energy, E C is the charging energy, and kz + b = Φ ext /Φ 0 with the weak external flux Φ ext (Φ ext depends linearly on the amplitude of the Z pulse z, with k and b being the slope and intercept, respectively). The RMD pulse consists of a series of square Z waves with duration of T. Owing to the constrained sampling rate of the DAC (digital-to-analogue converter), both the falling and rising edge durations are inherently limited to a minimum of about 0.5 ns. In our experiments, these square waves are substituted to trapezoidal waves with edges of 0.5 ns.
To further characterize the RMD pulse, we measure the dynamical phase induced during its operation. Initially, the qubit is prepared at its idle point and excited using an X/2 gate. Then, the qubit is biased to the working point using the RMD pulse and the flat pulse following a delay, respectively. After turning off all Z pulses to tune the qubit back to its idle point, we apply another rotation \({R}_{\phi }(\frac{{\rm{\pi }}}{2})\), with ϕ ranging from 0 to 2π. The population of \(| 1\rangle \) state reaches its maximum only when ϕ compensates for the accumulated dynamical phase. In Extended Data Fig. 1a,b, we show the accumulated phase in the RMD case
$${\varphi }_{{\rm{r}}}(t)={\int }_{0}^{t}{\rm{d}}t\,[{\omega }_{{\rm{r}}}(t)-{\omega }_{{\rm{i}}{\rm{d}}{\rm{l}}{\rm{e}}}],$$ (7)
where ω r (t) is the qubit frequency under RMD, and the phase in the flat case
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