Bosonic field digitization for quantum computers

Quantum simulation of quantum field theory is a flagship application of quantum computers that promises to deliver capabilities beyond classical computing. The realization of quantum advantage will require methods that can accurately predict error scaling as a function of the resolution and parameters of the model and that can be implemented efficiently on quantum hardware. In this paper, we address the representation of lattice bosonic fields in a discretized field amplitude basis, develop methods to predict error scaling, and present efficient qubit implementation strategies. A low-energy subspace of the bosonic Hilbert space, defined by a boson occupation number cutoff, can be represented with exponentially good accuracy by a low-energy subspace of a finite-size Hilbert space. The finite representation construction and the associated errors are directly related to the accuracy of the Nyquist-Shannon sampling and the finite Fourier transforms of the boson number states in the field and the conjugate-field bases. We analyze the relation between the boson mass, the discretization parameters used for wave function sampling, and the finite representation size. Numerical simulations of small size ${\mathrm{\Phi }}^4$ problems demonstrate that the boson mass optimizing the sampling of the ground state wave function is a good approximation to the optimal boson mass yielding the minimum low-energy subspace size. However, we find that accurate sampling of general wave functions does not necessarily result in accurate representation. We develop methods for validating and adjusting the discretization parameters to achieve more accurate simulations.

Figure 1

(a) Hermite-Gauss functions ${\varphi }_0\left(\phi \right), {\varphi }_{15}\left(\phi \right)$, and ${\varphi }_{34}\left(\phi \right)$ (respectively, solid, dashed, and dot-dashed lines) and the discrete harmonic oscillator (with $m_0=1$) eigenstates ${\tilde{\varphi }}_0\left({\phi }_i\right), {\tilde{\varphi }}_{15}\left({\phi }_i\right)$, and ${\tilde{\varphi }}_{34}\left({\phi }_i\right)$ (respectively, circle, square, and triangle symbols) for a finite Hilbert space with $N_{\phi }=64$ discretization points. Within $O\left({10}^{-4}\right)$ accuracy the support of the HG functions with $n\le N_b=34$ is inside the interval $[-L,L]$, where $L=\sqrt{\pi N_{\phi }/2}$ [see Eq. (29)]. These HG functions are sampled accurately by the discrete harmonic oscillator eigenvectors. (b) The support of ${\varphi }_{57}\left(\phi \right)$ (solid line) has significant weight outside $[-L,L]$, and the function cannot be sampled by the eigenvector ${\tilde{\varphi }}_{57}\left({\phi }_i\right)$ (circle symbols) of the discrete harmonic oscillator.

Figure 2

Within $O\left(\epsilon \right)$ accuracy, the algebra generated by the field operator $\mathrm{\Phi }$ and $\mathrm{\Pi }$ restricted to the $N_b$ size low-energy subspace of the harmonic oscillator Hamiltonian (1) (shaded region, left side) is isomorphic with the algebra generated by the discrete field operators $\tilde{\mathrm{\Phi }}$ and $\tilde{\mathrm{\Pi }}$ restricted to the the $N_b$ size low-energy subspace of the discrete harmonic oscillator Hamiltonian (45) (shaded region, right side). The accuracy increases exponentially with increasing the size $N_{\phi }$ of the finite Hilbert space; see Eq. (61).

Figure 3

Tail weight ${\epsilon }_w\left(n\right)$ [Eq. (30)] (solid lines) of HG functions vs $n$ for $N_{\phi }=64$ (upper, black), $N_{\phi }=128$ (middle, red) and $N_{\phi }=256$ (lower, blue) discretization points. Equation (54) (dashed lines) is a good approximation for small $n\lesssim 0.3N_{\phi }$. For larger $n, {\epsilon }_w\left(n\right)={\epsilon }_c(n-2)/\left(1.5\sqrt{n(n-1)}\right)$ with ${\epsilon }_c\left(n\right)$ given by Eq. (61) (dotted lines) provides a better bound for the error.

Figure 4

The tail weight, ${\epsilon }_w\left(n\right)$, Eq. (30), of HG functions (dash-dot black), the difference between the discretized HG and the discrete harmonic oscillator, ${\epsilon }_d\left(n\right)$, Eq. (55) (thin solid red), the error associated with the $\tilde{\mathrm{\Pi }}$ operator ${\epsilon }_{\mathrm{\Pi }}\left(n\right)$, Eq. (56) (dash-dot-dot green), the error associated with the $\tilde{\mathrm{\Phi }}\tilde{\mathrm{\Pi }}$ operator ${\epsilon }_{\mathrm{\Phi }\mathrm{\Pi }}\left(n\right)$, Eq. (57) (thick solid blue), and the commutation relation error, ${\epsilon }_c\left(n\right)$, Eq. (58) (orange circle symbols). In good approximation ${\epsilon }_d\left(n\right)\approx 1.5{\epsilon }_w\left(n\right)$ (dashed red line), ${\epsilon }_{\mathrm{\Pi }}\left(n\right)\approx \sqrt{n+1}{\epsilon }_d(n+1)$ (dotted green line), and ${\epsilon }_{\mathrm{\Phi }\mathrm{\Pi }}\left(n\right)\approx {\epsilon }_c\approx \sqrt{(n+1)(n+2)}{\epsilon }_d(n+2)$ (dot-dash-dash blue line). The size of the finite Hilbert space is (a) $N_{\phi }=64$ and (b) $N_{\phi }=128$.

Figure 5

(a) The square amplitude of the functions $f\left(\phi \right)$ [Eq. (73)] and $g\left(\phi \right)$ [Eq. (78)] (solid black) and of their Fourier transforms $\widehat{f}\left(\kappa \right)$ and $\widehat{g}\left(\kappa \right)$ (dashed blue), respectively. At this scale, $f$ and $g$ are indistinguishable. (b) ${\left|f\left(\phi \right)\right|}^2$ (solid black) and ${\left|g\left(\phi \right)\right|}^2$ (dashed green) for $\phi <-F$. (c) $|\widehat{f}{\left(\kappa \right)|}^2$ (solid blue) and $|\widehat{g}{\left(\kappa \right)|}^2$ (dashed red) for $\kappa <-K$. The weights of both $f$ and $g$ outside the sampling interval is small, $\approx O\left({10}^{-5}\right)$. (d) The function $s\left(\phi \right)$ [Eq. (79)] used to define $g\left(\phi \right)$. $s\left(\phi \right)$, and implicitly $g\left(\phi \right)$, decay exponentially fast at large $\left|\phi \right|$.

Figure 6

(a) Boson distribution of the wave functions $f\left(\phi \right)$ and $g\left(\phi \right)$ (solid black). Discrete harmonic oscillator eigenstate distribution of the discretized states $|\tilde{f}\rangle$ and $|\tilde{g}\rangle$ (hashed red). At large $n$ the boson and the discrete harmonic oscillator eigenstate distributions differ significantly. (b) The high-energy weight of the wave functions $f\left(\phi \right)$ and $g\left(\phi \right)$ vs the cutoff $N_b$. Fifty percent ($20\%$) of the wave functions' weight belongs to the high-energy subspace spanned by states with $n>30$ ($n>40$).

Figure 7

(a) $m_1$-boson distribution of the harmonic oscillator ground state for different values of the ratio $m_1/m_0$. (b) Boson cutoff number $N_b$ vs the high-energy subspace weight $1-W_{N_b}$ for different values of the squeezing parameter $r=\frac{1}{2}ln\left(\frac{m_1}{m_0}\right)$. (c) Boson cutoff number $N_b$ vs the squeezing parameter $r$ for different values of $1-W_{N_b}$. Numerical fitting yields $N_b\approx [-0.595-0.477ln(1-W_{N_b})]\frac{m_1}{m_0}$.

Figure 8

Local ${\varphi }^4$ field theory, Eq. (91), with $\frac{g}{m_0^3}=100$. (a) Field and conjugate-field distributions, ${\left|\mathrm{\Phi }\left(\phi \right)\right|}^2$ (dashed black) and $|\widehat{\mathrm{\Phi }}{\left(\kappa \right)|}^2$ (solid red), respectively, in the ground state. Insets: The conjugate-field distribution has an oscillatory behavior at large $\left|\kappa \right|$. (b) The ratio of the sampling intervals widths $\frac{K}{F}$ vs the tail weight $\epsilon$, calculated by employing Eq. (90). $\frac{K}{F}$ is larger than the bare mass $m_0$, and increases logarithmically with increasing the accuracy. (c) The number of the required discretization points $N_{\phi }=\lceil \frac{2}{\pi }FK\rceil$ increases logarithmically with decreasing the tail weight. (d) $m$-boson distribution for different choices of $\frac{m}{m_0}$. (e) The low-energy subspace cutoff $N_b$ vs $\frac{m}{m_0}$ for different truncation errors $1-W_{N_b}$. The optimal boson mass is found when $N_b$ is minimum.

Figure 9

Two-site ${\mathrm{\Phi }}^4$ field theory, Eq. (92), with $m_0^2=-1$ and $\frac{g}{|m_0{|}^3}=2$ and $h=1$ (a) Local field, Eq. (93) (dashed black) and conjugate-field, Eq. (94) (continuous red) distributions in the ground state. Inset: At large argument, the field distribution decays faster than the conjugate-field distribution. (b) The ratio of the sampling intervals widths $\frac{K}{F}$ vs the tail weight. $\frac{K}{F}$ is larger than $|m_0|$ and increases logarithmically with increasing accuracy. (c) The number of required discretization points $N_{\phi }=\lceil \frac{2}{\pi }FK\rceil$ increases logarithmically with decreasing the tail weight. (d) $m$-boson local distribution for different choices of $\frac{m}{|m_0|}$. (e) The low-energy subspace cutoff $N_b$ vs $\frac{m}{|m_0|}$ for different truncation errors $1-W_{N_b}$. The optimal boson mass is found when $N_b$ is minimum.

Figure 10

At every lattice site $n_q={log}_2\left(N_{\phi }\right)$ qubits are assigned to represent the field. (a) The field amplitude distribution at site $j$ can is obtained by direct measurement of the qubits assigned for the site $j$. (b) The conjugate-field amplitude distribution requires an inverse Fourier transform, ${\tilde{\mathcal{F}}}^{-1}$ [see Eq. (42)] at $j$ before measurement. (c) Quantum phase estimation algorithm for measuring the boson distribution at site $j$. An ancillary register of size $n_r=n_q+1$ is used to store the phase factors associated with the evolution of the system under the action of a local discrete harmonic oscillator Hamiltonian [Eq. (109)].