Efficient Quantum Pseudorandomness with Nearly Time-Independent Hamiltonian Dynamics

Quantum randomness is an essential key to understanding the dynamics of complex many-body systems and also a powerful tool for quantum engineering. However, exact realizations of quantum randomness take an extremely long time and are infeasible in many-body systems, leading to the notion of quantum pseudorandomness, also known as unitary designs. Here, to explore microscopic dynamics of generating quantum pseudorandomness in many-body systems, we provide new efficient constructions of unitary designs and propose a design Hamiltonian, a random Hamiltonian of which dynamics always forms a unitary design after a threshold time. The new constructions are based on the alternate applications of random potentials in the generalized position and momentum spaces, and we provide explicit quantum circuits generating quantum pseudorandomness significantly more efficient than previous ones. We then provide a design Hamiltonian in disordered systems with periodically changing spin-glass-type interactions. The design Hamiltonian generates quantum pseudorandomness in a constant time even in the system composed of a large number of spins. We also point out the close relationship between the design Hamiltonian and quantum chaos.

Popular Summary

Quantum randomness is the key to understanding intriguing phenomena, such as thermodynamics in isolated quantum systems, quantum chaos, and quantum black holes, and it is also a powerful tool in quantum engineering. Generating true quantum randomness in large systems, however, is expensive and cannot be done in a reasonable amount of time. An approximation, called quantum pseudorandomness, is therefore of crucial importance. So far, the generation of quantum pseudorandomness in natural systems with many particles has not been fully explored, resulting in a lack of a solid basis for studying quantum random phenomena. In this work, we show that quantum pseudorandomness spontaneously appears after a short time in quantum many-body systems under certain circumstances.

We do so by first providing a new method of generating quantum pseudorandomness, based on periodic changes of random interactions in the position space and of those in the momentum space. Extending this, we provide theoretical prescriptions for quantum circuits that can be used to generate quantum pseudorandomness significantly more efficiently than previously known approaches. We then show that the dynamics in certain quantum spin-glass systems quickly generates quantum pseudorandomness. The systems further turn out to be a class of quantum chaos.

Our results can be used in quantum engineering to implement large devices and to demonstrate quantum advantages in information processing. Our approach also has implications in fundamental physics because it bridges pseudorandomness in quantum information theory to random dynamics in quantum disordered systems, accelerating explorations of fascinating phenomena in complex quantum systems, such as quantum duality, from the quantum information point of view.

Figure 1

(a),(b) Examples of local permutation check problems for $t=4$ and $N=10$. In (a), $K_{I_1}=\{1111,0110,1000,0001\}$ is a permutation of $K_{I_1}^{\prime }=\{0110,0001,1111,1000\}$ (blue dashed boxes). However, $K_{I_2}$ is not a permutation of $K_{I_2}^{\prime }$ (red dash-dotted boxes). Hence, $K$ is an $\{I_1\}$-local but not an $\{I_2\}$-local permutation of $K^{\prime }$, also implying that $K$ is not a row permutation of $K^{\prime }$. In (b), $K$ is identical to $K^{\prime }$ except the columns in the blue dashed boxes and is a 2-local permutation of $K^{\prime }$. However, $K$ fails to be a 3-local permutation of $K^{\prime }$ due to the last column (see, e.g., $K_{I_2}$ and $K_{I_2}^{\prime }$). Panel (c) illustrates a relation between $\mathrm{RDC}(\mathcal{I})$ and an $\mathcal{I}$-local permutation check problem. As diagonal gates act on $I_1$, $I_2$, and $I_3$, we first check if $K$ is a $\{I_1,I_2,I_3\}$-local permutation of $K^{\prime }$. That is, we check the permutation relations between sets of rows in the red dash-dotted, green dotted, and blue dashed boxes. If $K$ is $\{I_1,I_2,I_3\}$-local but not a row permutation of $K^{\prime }$, then $⟨K,K^{\prime }|{\mathbb{E}}_{D^Z\sim \mathsf{RDC}(\mathcal{I})}[(D^Z{)}^{\otimes t,t}]|K,K^{\prime }⟩=1$.

Figure 2

Panel (a) depicts iterations of $\mathrm{RDC}(\mathcal{I})$ and the Hadamard transformation. Panel (b) shows $\mathrm{RDC}({\mathcal{I}}_2)$, where random diagonal two-qubit gates are applied onto all pairs. The circuit is called ${\mathrm{RDC}}_{\mathrm{disc}}^{(t)}({\mathcal{I}}_2)$ when each two-qubit gate is replaced with $({\mathrm{diag}}_Z\{1,e^{i{\phi }_1}\}\otimes {\mathrm{diag}}_Z\{1,e^{i{\phi }_2}\}){\mathrm{diag}}_Z\{1,1,1,e^{i\vartheta }\}$, where the phases ${\varphi }_1$, ${\varphi }_2$ and $\vartheta$ are chosen from discrete sets given in the main text.

Figure 3

Schematic figures illustrating the distributions of random unitaries in a whole unitary group. For the visualization, the unitary group is represented by an ellipse and each red dot corresponds to a unitary operator. Panel (a) illustrates a Haar random unitary, which is uniformly and continuously distributed over the whole unitary group. For unitary designs, the distribution is not necessarily continuous and is often defined on a finite support, which is depicted in (b). Panel (c) provides an intuitive picture of time-evolution operators generated by a design Hamiltonian, starting from the identity. As time passes, a design Hamiltonian generates random unitary distributed over the whole unitary. The time evolution is illustrated by a trajectory in the panel. When the design Hamiltonian is defined on a finite ensemble of Hamiltonians, there exists a time $T_{\mathrm{rec}}$, where all time evolution operators are in the neighborhood of the identity, due to the Poincaré recurrence theorem as depicted in (d).

Figure 4

A schematic figure about the design Hamiltonian $H_{XZ}{\in }_R{\mathfrak{H}}_{XZ}^{(t)}$. At each time interval $m$, $H_Z^{(m)}$ or $H_X^{(m)}$ is chosen uniformly at random from ${\mathfrak{H}}_Z^{(t)}$ or ${\mathfrak{H}}_X^{(t)}$, respectively. As depicted at the bottom of the figure, a random unitary generated by $H_{XZ}$ rapidly spreads over the whole unitary group and forms unitary designs in a short time independent of the system size.