Digitization of scalar fields for quantum computing

Qubit, operator, and gate resources required for the digitization of lattice $\lambda {\varphi }^4$ scalar field theories onto quantum computers are considered, building upon the foundational work by Jordan et al. [Quantum Inf. Comput. 14, 1014 (2014); Science 336, 1130 (2012)], with a focus towards noisy intermediate-scale quantum devices. The Nyquist-Shannon sampling theorem, introduced in this context by Macridin et al. [Phys. Rev. A 98, 042312 (2018)] building on the work of Somma [Quantum Inf. Comput. 16, 1125 (2016)], provides a guide with which to evaluate the efficacy of two field-space bases, the eigenstates of the field operator, as used by Jordan et al., and eigenstates of a harmonic oscillator, to describe ($0+1$)- and ($d+1$)-dimensional scalar field theory. We show how techniques associated with improved actions, which are heavily utilized in lattice QCD calculations to systematically reduce lattice-spacing artifacts, can be used to reduce the impact of the field digitization in $\lambda {\varphi }^4$, but are found to be inferior to a complete digitization improvement of the Hamiltonian using a quantum Fourier transform. When the Nyquist-Shannon sampling theorem is satisfied, digitization errors scale as $|log|log|{\epsilon }_{\mathrm{dig}}\left||\right|\sim n_Q$ (number of qubits describing the field at a given spatial site) for the low-lying states, leaving the familiar power-law lattice-spacing and finite-volume effects that scale as $|log|{\epsilon }_{\mathrm{latt}}||\sim N_Q$ (total number of qubits in the simulation). For localized (delocalized) field-space wave functions, it is found that $n_Q\sim 4\left(7\right)$ qubits per spatial lattice site are sufficient to reduce theoretical digitization errors below error contributions associated with approximation of the time-evolution operator and noisy implementation on near-term quantum devices. Only classical computing resources have been used to obtain the results presented in this work.

Figure 1

Identifying three distinct sources of error in quantum simulations of scalar field theory: 1, the error of digitizing and latticizing the continuous field onto qubit degrees of freedom; 2, the error of simulation due to the use of approximations to the exact digitized propagator; and 3, the error due to noise in the quantum device's implementation of the approximate digitized propagator. The first two sources of error are independent of the quantum hardware. The first is the main focus of this paper.

Figure 2

Precision of the energies of the five lowest states of the HO in Eq. (7) expected from calculations on an ideal quantum computer with JLP digitizations over a range of values of ${\overline{\varphi }}_{\max }$ for a system digitized on $n_Q=3$, 4, and 5 qubits (minimized at ${\overline{\varphi }}_{\max }=3.1$, 4.7, and 6.9, respectively). The vertical gray dashed lines correspond to saturation of the NS sampling bound.

Figure 3

Precision of the ground-state energy of the HO in Eq. (7) for unimproved, improved, and exact conjugate-momentum operators, over a range of digitizations of $\overline{\varphi }$ with different levels of gate noise. The light green points correspond to implementing the finite-difference conjugate-momentum operator, the light blue corresponds to the $O\left({\delta }_{\tilde{\varphi }}^2\right)$-improved conjugate-momentum operator, and the purple points correspond to the exact conjugate-momentum operator. Gaussian noise with a width $\sigma$ is added to the diagonal elements of the eigenvalues of the conjugate-momentum operators (resembling a simplified version of process 3 in Fig. 1). The maximum value of the field is fixed to be ${\overline{\varphi }}_{\max }=5.5$, enabling a precision of $\sim {10}^{-12}$ for an ideal quantum computer. The vertical light gray dashed lines correspond to the number of qubits associated with the number of states, while the darker gray solid line corresponds to the naive estimate of saturation of the NS sampling bound based upon the properties of the HO ground-state wave function (the six calculations cross the latter of these lines from top to bottom in the order of the legend). For the computational errors anticipated in the NISQ era, four qubits are seen to be sufficient to eliminate the digitization of the scalar field as a source of important error.

Figure 4

Precision of calculations of the ground-state energy of the HO in Eq. (7). (a) Expectations for an ideal quantum computer for different values of ${\overline{\varphi }}_{\max }$ as a function of the number of states. The vertical gray dashed lines correspond to the location of inflection points predicted by the NS sampling theorem for the indicated values of ${\overline{\varphi }}_{\max }$. A fit to the gray points calculated at the NS saturation point indicates $\epsilon \sim [1.8\left(2\right)\times {10}^3]\times 2^{-2.234\left(4\right)n_s}$, quantifying the double-exponential scaling between $\epsilon$ and $n_Q$. (b) Expected precision from a device with noise at the level of $\sigma ={10}^{-5}$ in the application of the field conjugate-momentum operator (as described in the text). In both panels, the NS saturation points arise left to right in the top-to-bottom order of the legend.

Figure 5

Expected precision of the ground-state energy of the HO Hamiltonian in Eq. (7) on an ideal quantum computer using a HO basis defined by ${\omega }_{\varphi }$ in Eq. (15) as a function of the number of basis states. The inset shows the scaling for a HO basis of 20 states, which is a slice of the main figure indicated by a vertical gray dashed line (the calculations cross this line from top to bottom in the order of the legend). In the limit of ${\omega }_{\varphi }=1$ the lowest-lying basis state is an eigenstate, and $\epsilon =0$. Tuning ${\omega }_{\varphi }$ to be in the vicinity of the optimal value ${\omega }_{\varphi }=1$ outperforms field-space digitization, shown by the gray JLP curve.

Figure 6

(a) Expected precision of calculations of the ground-state energy of a one-site $\lambda {\varphi }^4$ scalar field theory performed with an ideal quantum computer for different values of ${\overline{\varphi }}_{\max }$ as a function of the number of states (the NS saturation points arise left to right in the top-to-bottom order of the legend). (b) Field and (c) conjugate-momentum-space wave functions for the free (green or lightly shaded) and interacting (blue or darkly shaded) one-site $\lambda {\varphi }^4$ shown at constant ${\overline{\varphi }}_{\max }=2.5$. The introduction of nonzero $\lambda$ reduces the spatial support of the wave function while increasing its support in momentum space. The short-dashed green (lightly shaded) and blue (darkly shaded) lines in (c) are Fourier transforms of Gaussian fits to the wave functions in (b). The vertical gray dashed lines in (c) show the truncations in $\overline{\mathrm{\Pi }}$ for 6 and 18 states, the location of NS saturation for ${\overline{\varphi }}_{\max }=2.5$ as seen in Fig. 4 and (a) here.

Figure 7

Exploration of sensitivity in JLP field digitization (dashed lines) and the HO basis (solid lines) to tuning of digitization parameters determining the low-energy states in momentum space. For JLP field digitization, the relevant parameter is ${\overline{\varphi }}_{\max }$, while for the HO basis it is a combination of the frequency defining the basis ${\omega }_{\varphi }$ and the number of states $\sim \sqrt{2^{n_Q}/{\omega }_{\varphi }}$. The horizontal axes of the HO curves have been rescaled to $1.4\sqrt{2^{n_Q}/{\omega }_{\varphi }}$ to align them with the JLP curves. These tuning curve pairs are minimized with smaller values of $\epsilon$ in the top-to-bottom order of the legend.

Figure 8

Exploration of the sensitivity in JLP field digitization (dashed lines) and the HO basis (solid lines) to tuning of digitization parameters determining the low-energy states for a (0+1)-dimensional scalar field theory with $m^2<0$ and (a) $\mu =2$ and (b) $\mu =5$, resulting in delocalized wave functions. For JLP field digitization, a relevant parameter is ${\overline{\varphi }}_{\max }$, while for the HO basis it is a combination of the frequency defining the HO basis ${\omega }_{\varphi }$ and the number of states per site. The horizontal axes of the HO curves have been rescaled to (a) $2.9\sqrt{2^{n_Q}/{\omega }_{\varphi }}$ and (b) $1.8\sqrt{2^{n_Q}/{\omega }_{\varphi }}$ to align them with the JLP curves. The black dashed horizontal line in (a) corresponds to the precision required to distinguish between the ground state and first excited state for $\mu =2$. The corresponding line for $\mu =5$ in (b) lies many orders of magnitude beyond the range of the figure. In both panels, the tuning curve pairs are minimized with smaller values of $\epsilon$ in the top-to-bottom order of the legend.

Figure 9

(a) Potentials and (b) and (c) wave functions of the ground states (solid curves) and first excited states (dashed curves) for systems with $m^2<0$. (b) shows the spatial wave functions for $\lambda =1$ and $\mu =1,2$, while (c) shows the corresponding momentum-space wave functions. These wave functions result from using the JLP basis with ${\overline{\varphi }}_{\max }=9$ and $n_Q=7$ qubits. [The vertical dashed gray lines in (c) indicate the number of states at which the NS bound is saturated and thus an increase in ${\overline{\varphi }}_{\max }$ would be profitable over an increase in quantum states.]

Figure 10

Ground-state and first-excited-state wave functions for two-site $\lambda {\varphi }^4$ scalar field theory with $\lambda =0$ (top) and $\lambda =32$ (bottom). The first excited state shows positive correlation between the oscillations at the two spatial sites $({\overline{\varphi }}_0,{\overline{\varphi }}_1)$.

Figure 11

Precision of the calculated ground-state energy for the two-site lattice $\lambda {\varphi }^4$ scalar field theory with $\lambda =32$ performed with an ideal quantum computer for different numbers of qubits as a function of support in field space. For the JLP basis, the relevant parameter is ${\overline{\varphi }}_{\max }$, while for the HO basis it is a combination of the frequency defining the HO basis ${\omega }_{\varphi }$ and the number of states per site. The shown precision does not include deviations of this two-site ($1+1$)-dimensional theory from the continuum limit of the ($1+1$)-dimensional theory for which the number of spatial sites approaches infinity for a fixed spatial extent. The horizontal axes of the HO curves have been rescaled to $1.4\sqrt{2^{n_Q}/{\omega }_{\varphi }}$ to align them with the JLP curves. These tuning curve pairs are minimized with smaller values of $\epsilon$ in the top-to-bottom order of the legend.

Figure 12

Quantum circuit required to perform Trotterized time evolution of the HO Hamiltonian in Eq. (7), digitized with $n_Q=3$ qubits in JLP digitization, denoted by ${\tilde{H}}_3$. The upper circuit implements the operator ${\mathrm{\Phi }}_3\left(\theta \right)$ defined in Eq. (A6), with an arbitrary angle $\theta$, while the lower circuit implements that circuit in both $\tilde{\varphi }$ and $\tilde{\mathrm{\Pi }}$ space, making use of a symmetric QuFoTr and its inverse, to achieve Trotterized Hamiltonian evolution of the system defined by ${\tilde{H}}_3$. These circuits are equivalent to controlled-rotation gate circuits appearing in [34, 36].

Figure 13

Partial circuit for one first-order Trotterized step of time evolution for a single site of the interacting scalar field ${\overline{\varphi }}^2$ and ${\overline{\varphi }}^4$ terms with $n_Q=6$ in the JLP basis. Boxes indicate cnot pairs that can be eliminated, resulting in the reduced circuit in the second expression.

Figure 14

Quantum circuits required to perform time evolution of the HO Hamiltonian in Eq. (7) using a HO basis with $n_Q=3$. The upper circuit is applied when the HO basis can be precisely tuned to the HO being simulated. When detuned, the lower circuit is necessary with coefficients $c_{ijk}$ attained from the corresponding operators in Eq. (C2).

Figure 15

Visual representations of Eqs. (E1, E2, E3). From left to right, the finite-difference, ${\delta }_{\overline{\varphi }}^2$-improved, and exact field conjugate-momentum operators obtained from PBCs show increasing nonlocality in field space.

Figure 16

Visual representations of Eqs. (E4, E5, E6). From left to right, the finite-difference, ${\delta }_{\overline{\varphi }}^2$-improved, and exact field conjugate-momentum operators obtained from twisted boundary conditions show increasing nonlocality in field space.

Figure 17

Distance measure (Schatten 1-norm) between the noisily Trotterized and exactly digitized time-evolution operator (after steps 3 and 1 of Fig. 1, respectively) as a function of the Trotter step size $\delta t$ for an integrated evolution time of $T_f=1$. The noisy first-order Trotterized propagator ${\tilde{U}}_{\mathrm{Trot}}$ is implemented with sampled error rates on one- and two-qubit gates set by ${\sigma }_{\theta }$ and ${\sigma }_{cnot}$, respectively. The digitization scheme is defined by $n_Q=3$ and ${\overline{\varphi }}_{\max }=3.0$. From right to left, the calculations deviate from the ideal result in the top-to-bottom order of the legend with the last calculation at ${\sigma }_{cnot}={10}^{-8}$ maintaining visual agreement for the entire plotted domain.

Figure 18

Ground-state persistence of a one-site free scalar field propagated to time $T_f=10$ that has been digitized with $n_Q=3$ with ${\overline{\varphi }}_{\max }=3.0$ as a function of Trotter step size $\delta t$ for two error rates. The star-shaped points are calculated from a first-order Trotterized propagator without quantum noise and become exact in the limit $\delta t\to 0$. Points joined by colored lines are sampled with the noise model described in the text at error rates of (a) ${\sigma }_{cnot}={10}^{-3}$ and (b) ${\sigma }_{cnot}={10}^{-2}$ and ${\sigma }_{\theta }$ one order of magnitude smaller in each case. Eigenvector indices $n$ are indicated at the right of each panel.

Figure 19

Time evolution of the expectation value of the field operator for a state initialized to the ground state rotated by a single site in $\varphi$ space. (a) The evolution (gold curve) is the noiseless first-order Trotterization (after step 2 in Fig. 1) with $\delta t=0.3$. (b)–(d) The gate-error rate ${\sigma }_{cnot}$ increases to the right with ${\sigma }_{\theta }$ an order of magnitude smaller in each case. The deviation shown in (e)–(g) is with respect to the noiseless Trotterization and represents only the errors arising from step 3 of Fig. 1.