Quantum criticality using a superconducting quantum processor

Quantum criticality emerges from the collective behavior of many interacting quantum particles, often at the transition between different phases of matter. It is one of the cornerstones of condensed matter physics, which we access on noisy intermediate-scale (NISQ) quantum devices by leveraging a dynamically driven phenomenon. We probe the critical properties of the one-dimensional quantum Ising model on a programmable superconducting quantum chip via a Kibble-Zurek process, obtain scaling laws, and estimate critical exponents despite inherent sources of errors on the hardware. In addition, we investigate how the improvement of NISQ computers (more qubits, less noise) will consolidate the computation of those universal physical properties. A one-parameter noise model captures the effect of imperfections and reproduces the experimental data. Its systematic study reveals that the noise, analogously to temperature, induces a new length scale in the system. We introduce and successfully verify modified scaling laws, directly accounting for the noise without any prior knowledge. It makes data analyses for extracting physical properties transparent to noise. By understanding how imperfect quantum hardware modifies the genuine properties of quantum states of matter, we enhance the power of NISQ processors considerably for addressing quantum criticality and potentially other phenomena and algorithms.

Figure 1

(a) Quantum system dynamically driven from a point A (paramagnetic phase—PM, for the quantum Ising model considered here) to a point B (ferromagnetic phase—FM) of its phase diagram, with a transition happening on the way, characterized by a quantum critical point (QCP). (b) Decomposition of the operator ${\widehat{U}}_m^{zz}\left(\varphi \right)=exp\left(i\varphi {\widehat{Z}}_m{\widehat{Z}}_{m+1}\right)$ by sandwiching a single-qubit rotation gate around the $z$ axis by two-qubit CNOT gates. (c) Quantum circuit for the discretized unitary operation of Eq. (2) for the quantum Ising model Eq. (3). The PM ground state is constructed by applying Hadamard gates $H$ on individual qubits. The second step is the time-evolution, generated by a first-order Suzuki-Trotter expansion. Here, ${\widehat{U}}_m^x\left(\varphi \right)=exp\left(i\varphi {\widehat{X}}_m\right)$.

Figure 2

(a)–(c)–(e)–(g) Two-point correlation function of Eq. (4) at $t=0$ plotted versus the distance $x$ for different drive times $T$. (b)–(d)–(f)–(h) Rescaled two-point correlation function according to Eq. (5) with $\nu =z=1$ and $\eta =1/4$. (a), (b) Tensor network emulation of the quantum circuit for $L=257$ qubits with a second-order Suzuki-Trotter expansion and time step $\delta t=0.1$. (c), (d) Perfect emulation of the quantum circuit using $L=7$ qubits and performing two time steps of different duration $\delta t$ to access various drive times $T$. (e), (f) Simulation on Rigetti Aspen-9 superconducting quantum chip using the same parameters as (a) and (b). (g), (h) Noisy emulation of the quantum circuit to model the imperfect hardware. (i), (j) Chi-square per degree of freedom ${\chi }^2/N_{\mathrm{dof}}$ quantifying the quality of the data collapse for the two-point correlation function of Eq. (4) as a function of the critical exponents $\nu$ and $\eta$, see Supplemental Material [51] (smaller is better). The best collapse should be obtained from the genuine values of $\nu$ and $\eta$. The exact values are marked at the intersection of the two bold straight white lines. (i) Using the benchmark data of (a). (j) Using the quantum processor data of (e).

Figure 3

(a) Two-point correlation of Eq. (4) as a function of the distance $x$, rescaled by the $p=0$ data for $L=33$ qubits. Emulation details: $36513$ gates comprised of Hadamard, ${\widehat{U}}^x$, and ${\widehat{U}}^{zz}$ with $T=32, t=0$, and second-order Suzuki-Trotter expansion where $\delta t=0.1$. The results are averaged over $\gtrsim 2\times {10}^3$ random circuits. Each curve corresponds to a value of $p$ whose code color can be read from panel (b). Fit of the observed exponential decay $\sim exp\left[-x/\xi \left(p\right)\right]$ (bold line) to extract length $\xi \left(p\right)$. (b) Length $\xi \left(p\right)$ as a function of $p$, which shows a $\sim 1/p$ dependence. (c), (d) Rescaled two-point correlation function according to Eq. (5) ($z=\nu =1$ and $\eta =1/4$). The form of the circuit is the same as the one in panel (a) with $p={10}^{-4}$ for various drive times $T$. (d) Same as (c) except that the y-axis is multiplied by an additional term with $\tilde{\xi }\approx 180$, see Eq. (7).