Deep quantum circuit simulations of low-energy nuclear states

Abstract

Numerical simulation is an important method for verifying the quantum circuits used to simulate low-energy nuclear states. However, real-world applications of quantum computing for nuclear theory often generate deep quantum circuits that place demanding memory and processing requirements on conventional simulation methods. Here, we present advances in high-performance numerical simulations of deep quantum circuits to efficiently verify the accuracy of low-energy nuclear physics applications. Our approach employs novel methods for accelerating the numerical simulation including management of simulated mid-circuit measurements to verify projection based state preparation circuits. We test these methods across a variety of high-performance computing systems and our results show that circuits up to 21 qubits and more than 115,000,000 gates can be efficiently simulated.

Implementation of the Ge et al. algorithm

We briefly review here the algorithm used for state preparation that follows the work of Ref. [40] and describe how it has been implemented. We consider an Hamiltonian with spectrum in an interval $I\in (0,1)$, spectral gap $\Delta$, an initial trial state $|{\Phi } \rangle$ with overlap $\chi$ with the ground state $|{\Psi _0} \rangle$, and we assume we know the ground state energy E. After defining the shifted Hamiltonian $H^\prime =H-E\,\textbf{1}$ the state defined below

$$\begin{aligned} |{\tilde{\Phi }} \rangle =\frac{\cos ^M(H^\prime )}{||\cos ^M(H^\prime ) ||}|{\Phi } \rangle \end{aligned}$$

(13)

becomes close to the exact ground state, if the power of M is chosen as in Ref. [40]. It is possible to approximate the operator in Eq.  (13) as a linear combination of unitaries in the following way:

$$\begin{aligned} \cos ^{2m}H^\prime =\sum _{k=-m_0}^{m_0}\alpha _k e^{-2iHk}+R\,, \end{aligned}$$

(14)

with

$$\begin{aligned} \alpha _k=\frac{1}{4^m}\left( {\begin{array}{c}2m\\ m+k\end{array}}\right) \,. \end{aligned}$$

(15)

and $m_0$ properly chosen in order to vanish the quantity of R, a prescription is provided in Ref. [40]. The implementation of the above operator can be done using Linear Combination of Unitaries using the Prepare and Select oracles of Ref. [75] and briefly reviewed below. Given the $2m_0+1$ unitaries of Eq. 14 we define an ancillary register of dimension $n_A=\lceil \log _2(2m_0+1)\rceil$. We can implement a block encoding of the linear combination of unitaries using the algorithm originally developed in Ref. [76, 77] and using the approach reported in Ref. [78]. In particular we first need a prepare operator P acting only on the ancillary register defined by the following equation:

$$\begin{aligned} P|{0} \rangle= & {} \frac{1}{\sqrt{\alpha }}\sum _{k=0}^{2m_0}(\alpha _{-2m_0+k})^{1/2}|{k} \rangle \,, \end{aligned}$$

(16)

and $\alpha =\sum _{k=-m_0}^{m_0}|\alpha _k|$. Following Ref. [78] for the general case the gate decomposition of the unitary P can be done using generic circuit synthesis as originally reported in Ref. [79] whose exponential scaling should not be a limitation for the problems at hand (i. e. for $10^3$ unitaries only 10 ancillary qubits will be needed). After we define the select oracle, acting both on the ancillary register and the target register, in the following way

$$\begin{aligned} S= & {} \sum _{k=0}^{2m_0+1}|{k} \rangle \langle {k} |\otimes U^k\,, \end{aligned}$$

(17)

where $U=e^{-2iH}$. Although the implementetion in principle requires applying several multi-controlled unitaries, in Ref. [78], Lemma 3.5, has been shown that only a number of single controlled unitaries equal to the number of ancillary qubits are necessary and sufficient. Therefore the implementation of the Linear-combination-of-unitaries (LCU) method for the problem at hand can be expressed as in Fig. 2 of Ref. [78]. We are now ready to discuss the success probability of the procedure that is similar to Ref. [75]. Specifically, we define the following quantity:

$$\begin{aligned} \eta ^2= & {} \langle {\Phi } | O^2|{\Phi } \rangle \,, \end{aligned}$$

(18)

for which we have defined the operator $O=\sum _{k=-m_0}^{m_0}\alpha _k U^k$. The success probability of postselecting all 0s on the ancillary qubit is thus:

$$\begin{aligned} P_s= & {} \frac{\eta ^2}{\alpha ^2}\,. \end{aligned}$$

(19)