Emergent Quantum State Designs from Individual Many-Body Wave Functions

Quantum chaos in many-body systems provides a bridge between statistical and quantum physics with strong predictive power. This framework is valuable for analyzing properties of complex quantum systems such as energy spectra and the dynamics of thermalization. While contemporary methods in quantum chaos often rely on random ensembles of quantum states and Hamiltonians, this is not reflective of most real-world systems. In this paper, we introduce a new perspective: across a wide range of examples, a single nonrandom quantum state is shown to encode universal and highly random quantum state ensembles. We characterize these ensembles using the notion of quantum state $k$-designs from quantum information theory and investigate their universality using a combination of analytic and numerical techniques. In particular, we establish that $k$-designs emerge naturally from generic states in a Hilbert space as well as physical states associated with strongly interacting Hamiltonian dynamics. Our results offer a new approach for studying quantum chaos and provide a practical method for sampling approximately uniformly random states; the latter has wide-ranging applications in quantum information science from tomography to benchmarking.

Popular Summary

Generating random quantum states is a basic building block in quantum information science, with applications that include quantum supremacy tests, quantum cryptography, and quantum device verification. Existing methods require highly engineered, time-dependent control of quantum hardware and are only applicable to a narrow class of quantum systems that push the limits of today's technology. In our work, we show how to generate random quantum states without such fine-tuned controls and only require easily realized quantum evolution. We show that partial measurements on time-evolved quantum states generate the desired random quantum states. This enables many schemes requiring quantum randomness to be implemented on contemporary quantum systems and provides new insights on how quantum systems reach thermal equilibrium.

Our key innovation is to characterize how measurements on a subsystem of a single quantum state generate a large number of quantum states on the complementary subsystem, which we call the projected ensemble. We leverage statistical tools to study the randomness of this ensemble of states. A $k$-design is an ensemble of states that passes a certain test of randomness, formulated in terms of statistical moments. Our analytical and numerical findings show that the projected ensembles of easily preparable quantum states form $k$-designs.

Our results provide a new perspective on the long-standing heuristic that naturally occurring quantum dynamics mimics random behavior. The framework of projected ensembles provides a precise characterization of what it means for a nonrandom system to mimic randomness. Our approach allows for the characterization of nonequilibrium dynamics beyond conventional thermalization, a phenomenon which has come to be known as “deep thermalization.”

Figure 1

The emergence of a universal quantum state ensemble from a single many-body wave function. (a) A subsystem $B$ of a pure many-body wave function $|\mathrm{\Psi }⟩$ is measured in a fixed local basis; the remaining unmeasured qubits in $A$ are in a pure state that depends on the measurement outcome on $B$. (b) For quantum systems consisting of qubits, the measurement on $B$ samples random outcomes, each characterized as a bit string $z_B$ (binary numbers in the blue and yellow arrows). Different measurement outcomes $z_B$ occur with probability $p(z_B)$ and lead to distinct quantum states $|{\mathrm{\Psi }}_A(z_B)⟩$, forming the projected ensemble $\mathcal{E}=\{p(z_B),|{\mathrm{\Psi }}_A(z_B)⟩\}$. In the right panel, the ensemble of pure states $|{\mathrm{\Psi }}_A(z_B)⟩$ (black arrows) is randomly distributed in the Hilbert space of $A$ (black sphere), forming an approximate quantum state $k$-design. (c) Examples of many-body wave functions the projected ensembles of which form approximate quantum state designs include the typical output states of random unitary circuit evolution, quantum states obtained from quenched time evolution, and energy eigenstates of a chaotic Hamiltonian at infinite temperature. (d) An illustration of minimal quantum state $k$-designs for a single qubit on the Bloch sphere. The formation of higher $k$-designs requires a larger number of pairwise nonorthogonal quantum states. In the limit of $k\to \mathrm{\infty }$, a $k$-design approaches the so-called Haar ensemble, which is the uniform distribution over all pure quantum states.

Figure 2

A geometric arrangement of qubits for the solvable example of a generator state for an $\epsilon$-approximate $k$-design. The generator state can be efficiently prepared by time-independent Hamiltonian evolution. (a) A graph representing the connectivity of $N=N_r\times N_c$ qubits arranged in a two-dimensional square lattice. Qubits are represented by vertices (gray circles) and a pair of neighboring qubits interact via an Ising coupling whenever an edge (black line) exists between them. The connectivity exhibits a periodicity, where the two-row 12-column unit cell (blue box) is repeated in a brick-wall layout. (b) The connectivity and grouping of qubits within a single unit cell. Each qubit belongs to one of the four distinct groups as indicated by labels $\{a,b,c,d\}$.

Figure 3

Emergent quantum state designs from chaotic time evolution. (a) Quenched dynamics under a time-independent Hamiltonian $H$ starting from an initial product state, ${|0⟩}^{\otimes N}$, for a $N$-qubit system. The system is partitioned into two subsystems, $A$ and $B$, with sizes $N_A$ and $N_B$, respectively. At time $t$, a projective measurement in the local $z$ basis is performed on subsystem $B$, resulting in a specific outcome $z_B$ of length $N_B$. (b) The trace distances ${\mathrm{\Delta }}^{(k)}$ between the $k\mathrm{th}$ moments of the Haar ensemble and projected ensembles for an $N_A=3$ subsystem as a function of the evolution time for various total system sizes $N=12,14,\dots 24$ (darker colors for increasing $N$). The dashed lines are a phenomenological power-law fit, yielding a scaling of ${\mathrm{\Delta }}^{(k)}\sim t^{-1.2}$. (c) The design time ${\tau }_k$ defined by the evolution time to achieve a trace distance of ${\mathrm{\Delta }}^{(k)}=0.02$. We find that longer time evolution is required to form a higher approximate $k$-design. (d) Late-time trace distances at $t\approx {10}^3$. For a mixed-field chaotic Hamiltonian (circles), the late-time trace distances exhibit an exponential scaling with $N_B$ while they remain nearly constant for an integrable nonergodic Hamiltonian (crosses; time traces not shown). The dashed lines represent the trace distances ${\mathrm{\Delta }}_{\mathrm{em}}^{(k)}$ between the $k\mathrm{th}$ moments of the Haar ensemble and a finite set of $2^{N_B}$ states sampled from the Haar ensemble on $A$.

Figure 4

Emergent quantum state designs from energy eigenstates. (a) The trace distances between the $k\mathrm{th}$ moments of the Haar ensemble and a projected ensemble for an $N_A=3$ subsystem generated from the energy eigenstates of a Hamiltonian. The results are presented as a function of the energy density $E/N$ for a total system size of $N=14$. (b) The trace distances for the projected ensembles obtained from eigenstates near zero energy corresponding to infinite temperature. The distances exhibit an exponential decay as a function of the system size. The points are evaluated for 100 eigenstates near zero energy and the error bars denote their standard deviation. For comparison, the dashed lines represent the trace distances ${\mathrm{\Delta }}_{\mathrm{em}}^{(k)}$ between the Haar ensemble and an empirical ensemble of $2^{N_B}$ states sampled from the Haar ensemble.

Figure 5

A universal ensemble at finite temperature. (a) The trace distances between the second moments of the projected ensembles generated from a pair of energy eigenstates at $E_i$ and $E_j$ for $N_A=3$ subsystems. We plot the pairwise distances ${\mathrm{\Delta }}_{ij}^{(2)}$ for every pair of eigenstates $|E_i⟩,|E_j⟩$, computed for system size $N=11$. The distances are minimized when $E_i\approx E_j$. The plot suggests the existence of a universal ensemble that depends smoothly on the energy density. The black dashed line indicates a cut defined by $(E_i+E_j)/N=-0.6$. (b) The trace distances plotted as a function of the energy-density difference, $|E_i-E_j|/N$, along the black dashed line in (a) for various systems sizes $N=9,10,\dots ,15$ (darker colors for increasing $N$). The inset shows that the distances at zero energy difference $E_i=E_j$ decay exponentially with the system size. The trace distances from the projected ensemble are comparable to those from the finite-size ensemble, ${\mathrm{\Delta }}_{\mathrm{em}}^{(2)}$, of the empirical Haar distribution (dashed line). Such an exponential scaling suggests the existence of a universal random ensemble at finite temperatures, which we hypothesize to be the Scrooge ensemble. (c) The trace distance ${\mathrm{\Delta }}_E^{(2)}$ (dark red, dots) between the second moments of the projected ensemble and the corresponding Scrooge ensemble of eigenstates $|E⟩$, for a $N_A=3$ subsystem of a $N=14$ system. ${\mathrm{\Delta }}_E^{(2)}$ remains small for eigenstates $|E⟩$ in a wider energy range $E/N\in [-0.5,0.5]$, with an average value of $\overline{{\mathrm{\Delta }}^{(2)}}=0.11$ (dashed line). In contrast, the trace distance between the projected and Haar ensembles (light red, crosses) is strongly energy dependent and is large at energies away from infinite effective temperature $E=0$.

Figure 6

Emergent quantum state designs in Floquet dynamics. The trace distance ${\mathrm{\Delta }}^{(k)}$ versus the time for time evolution under the Floquet dynamics in Eq. (C1), for $k=1,2,3,4$ in (a), (b), (c), and (d), respectively. We plot the trace distance ${\mathrm{\Delta }}^{(k)}$ between the $k\mathrm{th}$ moments of the Haar ensemble and the projected ensembles for an $N_A=3$ subsystem and the total system sizes $N=12,14,\dots ,22$ (darker colors denote increasing $N$). The solid lines denote the Floquet dynamics, while the dotted lines denote the time-independent dynamics. The dashed line denotes the power-law fit for time-independent dynamics. The Floquet dynamics quickly saturate to their late-time ${\mathrm{\Delta }}^{(k)}$ value and deviate from power-law behavior. This suggests that the power-law decay of ${\mathrm{\Delta }}^{(k)}$ is associated with hydrodynamic behavior due to energy conservation.

Figure 7

Emergent quantum state designs in random-coupling and random-hopping models. (a),(b) The time-evolution (a) and eigenstate (b) data for the random all-to-all coupling model in Eq. (C2). The trace distances ${\mathrm{\Delta }}^{(k)}$ for $k=1$, $2$, and $3$ are plotted in orange, red, and purple, respectively, for $N_A=3$ subsystems [(a), left]. The time-evolution data for $N=21$ are plotted, along with the system-size scaling of the late-time ${\mathrm{\Delta }}^{(k)}$ for $N=8$ to $21$ [(a), right]. We also plot ${\mathrm{\Delta }}^{(k)}$ for each eigenstate for $N=14$ [(b), left], along with the system-size scaling of the eigenstate data near $E=0$ for $N=10$ to $14$ [(b), right]. (c),(d) The time-evolution (c) and eigenstate (d) data for the random-hopping model in Eq. (C3). Data are presented for $N_A=5$ subsystems, in the $S_{\text{tot.}}^z=0$ sector. In (c), time-evolution data for $N=24$ are plotted along with system-size scaling for $N=10,12,\dots ,24$. To illustrate the failure of projecting onto all strings $z_B$, we plot the results of this naive procedure with $N_A=3$ in light colors and dashes. In (d), eigenstate data for $N=16$ are plotted, along with their system-size scaling for $N=12,14,16$ and $18$. The system-size scaling is plotted against the effective $N_B$, which is ${\log }_2$ of the number of strings $z_B$ and which is postselected. In each plot, the error bars are smaller than the marker sizes.

Figure 8

A visualization of the modulating function and the concentration of $d_B⟨{\tilde{\mathrm{\Phi }}}_z|{\tilde{\mathrm{\Phi }}}_z⟩$. (a) The modulating function ${\mu }_{k,b}(s)$ defined in Eq. (E10) for varying values of $b$ from $2$ to $16$. The darker colors correspond to higher $b$. We fix $k=2$ in the plot. (b) The concentration of $s=d_B⟨{\tilde{\mathrm{\Phi }}}_z|{\tilde{\mathrm{\Phi }}}_z⟩$ for the Hilbert-space dimension of subsystem $A$, i.e., $d_A=2^{N_A}=16$, when $|\mathrm{\Phi }⟩$ is sampled uniformly at random from the complex sphere. We sample $10000$ different vectors $|\mathrm{\Phi }⟩$ to generate the histogram. The black solid line is a kernel-density-estimation fit of the histogram. If $d_A$ is larger, then $s$ is more concentrated.