Capacity and Quantum Geometry of Parametrized Quantum Circuits

To harness the potential of noisy intermediate-scale quantum devices, it is paramount to find the best type of circuits to run hybrid quantum-classical algorithms. Key candidates are parametrized quantum circuits that can be effectively implemented on current devices. Here, we evaluate the capacity and trainability of these circuits using the geometric structure of the parameter space via the effective quantum dimension, which reveals the expressive power of circuits in general as well as of particular initialization strategies. We assess the expressive power of various popular circuit types and find striking differences depending on the type of entangling gates used. Particular circuits are characterized by scaling laws in their expressiveness. We identify a transition in the quantum geometry of the parameter space, which leads to a decay of the quantum natural gradient for deep circuits. For shallow circuits, the quantum natural gradient can be orders of magnitude larger in value compared to the regular gradient; however, both of them can suffer from vanishing gradients. By tuning a fixed set of circuit parameters to randomized ones, we find a region where the circuit is expressive but does not suffer from barren plateaus, hinting at a good way to initialize circuits. We show an algorithm that prunes redundant parameters of a circuit without affecting its effective dimension. Our results enhance the understanding of parametrized quantum circuits and can be immediately applied to improve variational quantum algorithms.

Popular Summary

Quantum computers have the potential to tackle challenging problems in a variety of scientific areas. Quantum computers that are available now are utilized for applications such as variational quantum algorithms, quantum machine learning, and quantum sensing. These applications are frequently executed with parametrized quantum circuits (PQCs), which are composed of layers of parametrized unitaries. To harness the potential of quantum computers, these PQCs have to be able to express a wide range of quantum states.

This work introduces quantum geometric measures to evaluate and improve the capability of PQCs to express quantum states. We evaluate commonly used circuit types and find striking differences in their expressiveness. For deep circuits, we reveal a transition in the quantum geometry that indicates when a PQC has reached maximal expressiveness. Further, we propose better initialization strategies that are both expressive and trainable. Finally, we demonstrate an algorithm that removes redundant parameters of PQCs to reduce the amount of quantum resources needed to run quantum algorithms.

Using ideas from quantum geometry, our study provides new ways to evaluate the power of variational algorithms. Our findings pave the way for the design of improved quantum circuits and better initialization techniques for their usage.

Figure 1

(a) A sketch of a hardware-efficient parametrized quantum circuit (PQC) $U(\theta )|0{⟩}^{\otimes N}=\prod _{l=p}^1\left[W_l{V}_l({\theta }_l)\right]{\sqrt{H_d}}^{\otimes N}|0{⟩}^{\otimes N}$, with parameters $\theta$ and the initial state $|0⟩$ of all $N$ qubits being in state zero. The PQC consists of an initial layer of $\sqrt{H_d}$ gates applied to each qubit, where $H_d$ is the Hadamard gate, followed by $p$ repeated layers of parametrized single-qubit rotations $V_l({\theta }_l)$ and entangling gates $W_l$. $V_l({\theta }_l)$ consists of single-qubit rotations $R_{\alpha }({\theta }_{l,n})=\exp (-i{\sigma }_n^{\alpha }{\theta }_{l,n}/2)$ at layer $l$ and qubit $n$ around axis $\alpha \in \{x,y,z\}$. (b) The two-qubit entangling gates $w_l$ considered are controlled-not (cnot) gates (control-${\sigma }_x$), controlled-phase (cphase) gates (control-${\sigma }_z$, $\mathrm{diag}(1,1,1,-1)$), or $\sqrt{\mathrm{i}\mathrm{swap}}$ gates. (c) The entangling layer $W_l$ is composed of the two-qubit entangling gates $w_l$, which are arranged in a nearest-neighbor one-dimensional chain topology (denoted as chain), via an all-to-all connection (all) or in an alternating fashion (alt) for even and odd layers $l$.

Figure 2

An example to demonstrate the effective quantum dimension $G_C$ and the parameter dimension $D_C$ for a single qubit parametrized as $|\psi (\theta ,\phi )⟩=\cos (\theta /2)|0⟩+\exp (i\phi )\sin (\theta /2)|1⟩$. The quantum state is described by $M=2$ parameters $\theta$ and $\phi$. $D_C=2$ is the number of independent parameters of the quantum state. $G_C(\theta ,\phi )$ denotes the number of independent directions the quantum state can change to by locally perturbing its parameters $\theta$, $\phi$. For a random state $|v⟩$ ($\theta \notin \{0,\pi \}$) two possible directions exist, along $v_{\theta }$ and $v_{\phi }$. The particular state $|w(\theta =\pi ,\phi )⟩$ can only be perturbed in direction $w_{\theta }$, as adjusting $\phi$ does not change the state (e.g., $|w(\pi ,\phi +ϵ)⟩=|w(\pi ,\phi )⟩)$; thus $G_C(\theta =\pi ,\phi )=1$.

Figure 3

The properties of a PQC consisting of $p$ layers of randomly chosen $x$, $y$, and $z$ rotations, followed by cnot gates in a chain topology (see Fig. 1) for $N=10$ qubits. (a) The parameter dimension $D_C$ of the PQC [determined via Eq. (13)] scales linearly with $p$, until it levels at a characteristic value $p_c\approx 210$. (b) The variance of the logarithm of the nonzero eigenvalues of $\mathcal{F}$. The variance peaks around $p\approx p_c$. (c) The minimal nonzero eigenvalue of $\mathcal{F}$ against $p$. It increases for $p>p_c$. (d) The histogram of the logarithm of the eigenvalues of the Fisher information matrix $\mathcal{F}$. The width of the distribution increases with $p$, with a pronounced tail at small $\mathcal{F}$ developing around $p\approx p_c$, which disappears for $p>p_c$. (e) The variance of the gradient $\mathrm{var}({\mathrm{\partial }}_{k}E)$ and the QNG $\mathrm{var}({\mathcal{F}}^{-1}{\mathrm{\partial }}_{k}E)$ in respect to the Hamiltonian $H={\sigma }_1^z{\sigma }_2^z$. The gradient decays until $p\approx 20$, after which it remains constant. The QNG remains larger than the regular gradient but decreases for $p>p_c$. (f) The variance of the gradients and the QNG for varying qubit number $N$ for depth $p=2N$, showing an approximate exponential decrease with $N$.

Figure 4

The variance of the gradient and the redundancy of different hardware-efficient PQCs plotted against the number of layers $p$ for $N=10$ qubits. Each layer consists of single-qubit rotations as randomly chosen rotations around the $x,y,\text{ or }z$ axis. We plot different arrangements of entangling gates (as shown in Fig. 1 with (a),(b) a nearest-neighbor one-dimensional chain, (c),(d) all-to-all, and (e),(f) alternating nearest-neighbor connections. (a),(c),(e) The variance of the gradient $\mathrm{var}({\mathrm{\partial }}_{k}E)$ in respect to the Hamiltonian $H={\sigma }_1^z{\sigma }_2^z$. (b),(d),(f) The redundancy $R$ [Eq. (3)], which is the fraction of redundant parameters of the PQC.

Figure 5

The capacity of PQCs with $y$ rotations and different entangling gates. The entangling layer is arranged as a nearest-neighbor one-dimensional chain. Three of the PQCs have $y$ rotations and as reference we show a PQC with randomized $x$, $y$, or $z$ rotations and cnot gates. (a) The variance of the gradient $\mathrm{var}({\mathrm{\partial }}_{k}E)$ in respect to the Hamiltonian $H={\sigma }_1^z{\sigma }_2^z$. (b) The maximal parameter dimension $D_C$ of the PQCs as function of the number of qubits $N$. For $y$ cphase, we find an approximate power law $D_C\propto N^2$.

Figure 6

Tuning the PQC parameters $\theta =a{\theta }_{\mathrm{random}}$, where $a=(0,1]$ and ${\theta }_{\mathrm{random}}\in [0,2\pi )$ for circuits composed of random $x$, $y$, and $z$ rotations and entangling gates arranged in a chain configurations. (a) The effective quantum dimension $G_C$ as a function of ${\log }_{10}(a)$. The black dashed-dotted line is the number of parameters $M$. (b) The variance of the gradient $\mathrm{var}({\mathrm{\partial }}_{k}E)$ in respect to Hamiltonian $H={\sigma }_1^z{\sigma }_2^z$. All plots show the number of layers $p=100$ and $N=10$ qubits.

Figure 7

The effective quantum dimension $G_C(\theta =0)$ plotted against the number of qubits $N$ for a circuit consisting of randomly chosen parametrized rotations around the $x$, $y$, or $z$ axis with parameters $\theta =0$, and two-qubit entangling gates arranged in a nearest-neighbor chain. We compare cnot, cphase, and $\sqrt{\mathrm{i}\mathrm{swap}}$ entangling gates. From the numerical results, we find that $G_C$ scales quadratically for cnot gates, linearly for cphase gates, and with higher-order polynomial or even exponential scaling for $\sqrt{\mathrm{i}\mathrm{swap}}$. The number of layers $p$ is chosen such that $G_C(\theta =0)$ is maximized.

Figure 8

The properties of further PQCs plotted against layers $p$ for $N=10$ qubits. The PQCs have nearest-neighbor chain entangling layers. We define the types of PQCs in the legend: rand($xyz$) denotes randomized single-qubit rotations around the $x,y,\text{ and }z$ axes. By rand($xyw$), we denote randomized single-qubit rotations around the $x,y,\text{ and }(x+y)/\sqrt{2}$ axes. $zxz$ denotes that for every layer there are three single-qubit rotations, around the $z$, $x$, and $z$ axes. (a) The variance of the gradient $\mathrm{var}({\mathrm{\partial }}_{k}E)$ in respect to the Hamiltonian $H={\sigma }_1^z{\sigma }_2^z$. (b) The variance of the QNG $\mathrm{var}({\mathcal{F}}^{-1}{\mathrm{\partial }}_{k}E)$. (c) The redundancy $R$ of the parameters of the PQCs. (d) The variance of the logarithm of the eigenvalues of the QFI.

Figure 9

The variance of the gradient for $H={\sigma }_1^z{\sigma }_2^z$ (solid curves) and transverse Ising Hamiltonian (dashed curves) for different types of PQC with entangling gates in a chain configuration for a varying number of layers $p$. For the transverse Ising Hamiltonian, we divide the variance of the gradient by the number of terms in the Hamiltonian. We use $N=10$ qubits.

Figure 10

The distribution of eigenvalues of QFI for PQCs shown in Fig. 4 in the main text. (a) The nearest-neighbor chain arrangement of entangling gates, (b) All-to-all connectivity. (c) Alternating nearest-neighbor connections. All graphs for $N=10$ qubits and number of layers $p=50$.

Figure 11

The pruning of the layered PQC composed of random parametrized rotations with Pauli operators $\{{\sigma }^x,{\sigma }^y,{\sigma }^z\}$ and cphase gates arranged in a nearest-neighbor chain. The PQC has $p=40$ layers and $N=6$ qubits. The color scheme indicates the type of parametrized rotation at a specific qubit and layer. The identity operation $I$ is to show that no operation (identity) is applied at that particular qubit and depth, which results in pruning of the redundant rotations. (a) We show the initial PQC with $N_{\mathrm{param}}=240$ parameters and parameter dimension $D_C=126$. (b) Algorithm 1 removes redundant parametrized rotations and replaces them with identity operator $I$. We show the pruned PQC with $N_{\mathrm{param}}=126$ and the same parameter dimension $D_C=126$, reducing the PQC by 14 circuit layers and 114 parameters.

Figure 12

The Hadamard test to evaluate overlap $\mathrm{Re}[⟨{\mathrm{\partial }}_k\psi |{\mathrm{\partial }}_l\psi ⟩]$ for QFI.