Low overhead universality and quantum supremacy using only $Z$ control

We consider a model of quantum computation we call “varying $Z$” ($\mathrm{V}Z$), defined by applying controllable $Z$-diagonal Hamiltonians in the presence of a uniform and constant external $X$ field, and prove that it is universal, even in one dimension. Universality is demonstrated by construction of a universal gate set with $O\left(1\right)$ depth overhead. We then use this construction to describe a circuit whose output distribution cannot be classically simulated unless the polynomial hierarchy collapses, with the goal of providing a low-resource method of demonstrating quantum supremacy. The $\mathrm{V}Z$ model can achieve quantum supremacy in $O\left(n\right)$ depth in one dimension, equivalent to the random circuit sampling models despite a higher degree of homogeneity: it requires no individually addressed $X$ control.

Figure 1

(a) Example effective single-qubit gate layer $G_L$ in 1D, with a gate $g$ acting on only the first qubit. (b) Implementation as applied layers in the $\mathrm{V}Z$ model. The time $t$ is chosen to make the net effect on qubits $\{2,\cdots ,n\}$ the identity gate.

Figure 2

(a) $U_i$ depicted by its rotational axis in the $X\text{−}Z$ plane of the Bloch sphere. Here $c$ is picked to make the magnitude of rotation $2\pi +\gamma$ for qubits with $v_i=1$, which results in the rotational axis making an angle $\alpha$ with the $|0\rangle$ state. (b) The unitary $V$ has $t^{\prime }$ and $c^{\prime }$ such that the magnitude of rotation is ${\alpha }^{\prime }$ and the axis makes an angle $\psi$ with the $|0\rangle$ state for all qubits. (c) The net result of all three rotations $V{U}_i{V}^{†}$ is that the rotational axis of $U_i$ (dashed orange) is rotated by $V$ (blue) to point along $\vec{r}$ (solid orange) with spherical coordinates ($\theta ,\varphi$). The net rotation is of magnitude $2\pi +\gamma$ about the axis $\vec{r}$.

Figure 3

(a) Example effective two-qubit gate layer $G_L$ with a gate $g$ acting on only the first two qubits. (b) Its implementation as applied layers in the $\mathrm{V}Z$ model. Here $V=e^{-ia{t}^{\prime }X}$ is applied both before and after the coupling interaction, as in Eq. (15).

Figure 4

(a) An example of a single effective layer $L$ of an arbitrary 1D circuit. The circuit is composed of gates $g_l$ of the UGS $\mathcal{G}$, which act on at most two neighboring qubits. (b) The layer $L$ implemented as three effective sublayers. Each sublayer $l$ can be written as a tensor product of $g_l$ or the identity across all qubits, allowing it to be implemented in the $\mathrm{V}Z$ model in constant depth using the methods of this section.

Figure 5

(a) An example of an effective coupling sublayer $L$ of a 1D circuit, in which the coupling gate $g$ is applied across all qubits, rather than restricted to pairwise coupling. (b) The sublayer $L$ implemented as two effective coupling sublayers, each with pairwise coupling.

Figure 6

The $n=4$ case of a 1D $O\left(n\right)$-depth circuit capable of generating the distribution in Eq. (21) via alternating layers of $ZZ$ coupling and swap gates. It is equivalent to that of Ref. [14], but composed of Hadamard gates ($W$), swap gates, and $Z$-diagonal gates of the form $e^{i{v}_i{Z}_i}$ and $e^{i{w}_{ij}{Z}_i{Z}_j}$, denoted by their degree of rotation $v_i$ or $w_{ij}$. In this design qubits effectively act as particles moving past each other in 1D, and after $n$ layers each qubit has a chance to interact with each other qubit once. As $Z$-diagonal gates commute, multiple interactions between any pair of qubits can be implemented as a single interaction. Thus, this 1D depth-$O\left(n\right)$ circuit is sufficient to implement the circuit implied by Eq. (21).

Figure 7

(a) The first two layers of an example circuit with arbitrarily chosen $w_{ij}$ and $v_i$. $Z$-diagonal gates are denoted by their degree of rotation. (b) The same two layers rewritten as effective sublayers which can be implemented in the $\mathrm{V}Z$ model. For example, the gate corresponding to $v_2=\frac{5\pi }{8}$ in (a) is decomposed into $\frac{\pi }{2}$ and $\frac{\pi }{8}$ gates in (b), with the $\frac{\pi }{8}$ gate shifted so that it is executed in parallel with the other $\frac{\pi }{8}$ gate from (a) to form a sublayer.

Figure 8

The swap gate represented as (a) a product of cnot gates, (b) Hadamard ($W$) and CZ gates, and (c) Hadamard, $e^{-i\frac{\pi }{4}Z}$, and $e^{-i\frac{\pi }{4}Z\otimes Z}$ rotation gates. The latter may be implemented in the $\mathrm{V}Z$ model using the methods of Sec. 3.

Figure 9

Homogeneous vs alternating pattern of $X$ fields. Blue lines represent $ZZ$ couplings, which form a sublattice. Gray circles represent nonzero $X$ field, while white circles have no $X$ field.

Figure 10

Values of $C$ and $k$ for which Eq. (F5) can be solved for $x>0$. Color is included for visibility. For each value of $C$, at least one value of $k$ leads to a solution.