On Quantum Simulation Of Cosmic Inflation

The inflationary spacetime is given by the following metric

Here, $\bf{x}$ is the coordinate as the three-vector. $\tau$ is called the conformal time, where the metric is written in the conformally flat form. The scale factor $a(\tau)$ is given by

where $H$ is the Hubble parameter¹¹1We abuse the notation here, where $H$ is the Hubble constant, not the Hamiltonian., $a_{0}$ is the initial scale factor with the initial time $t_{0}$. The conformal time $\tau$ is related to the physical time $t$ by

At the beginning of inflation, we have $\tau_{0}\sim-\infty$, while at the end, we have $\tau_{\text{end}}\sim 0^{-}$. Thus, at the beginning of the inflation, which corresponds to the big bang, we have $a(\tau_{0})\sim 0$. In the end, we have a scale factor, which is very large. We often use the $e$-folding number

to measure the time of inflation.

Now, we consider a massless scalar field $\sigma$ moving in this background, with the action

where $\partial_{i}$ denotes the spatial derivative. Furthermore, we define $A^{\prime}=\partial_{\tau}A$ and $\dot{A}=\partial_{t}A$ for variable $A$. Since it is in the free theory, it could be exactly solved. The solution of the field operator in the Heisenberg picture reads

Here, $a$ and $a^{\dagger}$ are creation and annihilation operators

$u$ satisfies the classical equation of motion and $k\equiv\absolutevalue{\bf{k}}$²²2We apologize that we abuse the notation such that $\int d^{3}k$ means the integration over the three-vector $\bf{k}$. Similar definition works for $x$ and $\bf{x}$. We will also use $k_{i}$ to denote the $i$-th component $k_{i}=\mathbf{k}\cdot\mathbf{r}_{i}$, where $\mathbf{r}_{i}$ is the $i$-th unit vector.. The definition of the mode function $u$, and the definition of the vacuum, have some ambiguities due to the nature of the curved spacetime. A canonical choice is so-called the Bunch-Davies vacuum Bunch:1978yq ,

In the curved spacetime, the quantization defined by the creation and annihilation operators are analogous to the flat-space answer: the mode $\bf{k}$ is created from the creation operator $a^{\dagger}_{\bf{k}}$.

How is this related to the cosmic perturbation theory? In fact, the free field $\sigma$ here we are considering could be understood as the fluctuation of the inflaton field $\delta\phi(t,\bf{x})$, where

Here, $\bar{\phi}(t)$ is the homogenous and isotropic classical background that is not related to the coordinate, and $\delta\phi$ is understood as a quantum fluctuation (an operator) following the rules of quantum field theories. A similar treatment is done for computing thermal radiations of the black hole, where we are also expanding fluctuations around classical curved geometries.

In principle, cosmic perturbation theory requires a full consideration by expanding the metric, together with the inflationary field as well. Then, from the full expansion, one could decompose perturbations into different components based on the spin. In geometry, the curvature perturbation will come together with the inflaton, and there are ambiguities made by redefinitions of coordinates and fields. The redundancy could be fixed by the gauge choice. A standard and convenient choice is to replace the role of $\delta\phi$ by another variable $\zeta$, which represents the curvature perturbation, and this gauge is called the $\zeta$-gauge. Under the $\zeta$-gauge, we could write down the second-order action³³3We use the unit such that we set the Planck mass $M_{\text{pl}}=1$.

where $\epsilon$ is so-called the slow-roll parameter

and $V=V(\bar{\phi})$ is the inflationary potential⁴⁴4Roughly speaking, one could interpret $\zeta$ as $\zeta=-H\delta\phi/\dot{\bar{\phi}}$..

More general single field inflationary models have the following second-order action in the $\zeta$-gauge,

where $c_{s}$ is called the sound speed Chen:2006nt ⁵⁵5Defining the Lagrangian density $\displaystyle L\supset\sqrt{-g}P(\phi,X)~{},$ (14) for the whole inflaton field before perturbation, where $X\equiv-(\partial_{\mu}\phi\partial^{\mu}\phi)/2$, we could define the sound speed by $\displaystyle c_{s}^{2}=\frac{P_{,X}}{P_{,X}+2XP_{,XX}}~{}.$ (15) . In this case, the mode expansion reads

where

This action is exactly the free theory piece of the whole action we wish to simulate as the simplest example.

Now, motivated by the goal of quantum simulation, we define our theory in the lattice. The lattice version in this paper means that we are discretizing the spatial direction, replacing

where $b$ is the lattice spacing, $L$ is the total length of a single direction, and

The total number of sites is then given by

The notation $\mathbb{Z}_{\hat{L}}^{3}$ denotes the three-dimensional integer periodic lattice with the periodicity $\hat{L}$ in each direction. One could define a lattice version of the scalar quantum field theory based on the action of the curvature perturbation. From the Lagrangian density in the continuum,

we could write down a discrete version of the Lagrangian

where

is defined as the discrete version of the spatial derivative, and $\mathbf{r}_{i}$ is the unit vector along the spatial direction $i$. Note that the definition of the Lagrangian depends on the time coordinate, and here we are using the conformal time $\tau$.

Now we could perform the Legendre transform to define the Hamiltonian. The field momentum is

Thus, the Hamiltonian density in the continuum is

Thus, the lattice version is

Now we could write down the rule of canonical quantization

Now we make some brief comments on the applicability of the above theory. In our theory, we choose the ultraviolet cutoff $\Lambda=1/b$, corresponds to the spatial lattice spacing $b$. In cosmic inflation, we usually demand,

The right-hand side is concerning that we are avoiding Planckian physics, and we set the Planck constant to be 1 in our units.

Now, we extend our mode expansions to the lattice version,

where $\Gamma_{3}$ is the dual lattice

and

Note that in the formula, we assume that $v(-\mathbf{k})=v(\mathbf{k})$ to make sure we have a real field. $v(\mathbf{k})$ should satisfy the equation of motion for the corresponding classical discrete field theory. We find

When we compare this result with the continuous mode function, it is easy to realize the following properties of the solution.

• •

  We indeed have $v(-\mathbf{k})=v(\mathbf{k})$.

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  The form of the solution is intuitively understandable. The form of the dispersion relation is already presented in the case of flat space in the lattice theory Jordan:2011ne ; Jordan:2011ci . Since the metric is conformally flat, the dispersion relation should stay the same in our coordinate. Using the dispersion relation $\omega(\mathbf{k})$ in the equation of motion for the Fourier space, we will obtain $v(\mathbf{k})$.

• •

  In the limit where $\tau\to-\infty$, the mode function has reduced to the flat space lattice solution with the sound speed

  $\displaystyle v({\bf{k}})=\frac{{iH\tau\sqrt{{c_{s}}}}}{{2\sqrt{\epsilon\omega({\bf{k}})}}}{e^{-i\omega({\bf{k}}){c_{s}}\tau}}\Rightarrow\left|{\sqrt{2\epsilon}av({\bf{k}})}\right|=\frac{{\sqrt{{c_{s}}}}}{{\sqrt{2\omega({\bf{k}})}}}{e^{-i\omega({\bf{k}}){c_{s}}\tau}}~{}.$

  (33)

  Note that $\absolutevalue{\sqrt{2\epsilon}av(\mathbf{k})}$ the honest scalar field defined consistently for the flat space ⁶⁶6The coefficient $\sqrt{2\epsilon}$ comes from the relation between the curvature perturbation and the scalar field perturbation.. This is indeed the mode function of the “phonons” in the 3+1 dimensions. Thus, we indeed define a lattice version of the Bunch-Davies vacuum.

• •

  We could also look at the continuum limit $b\to 0$, where the solution is reduced to the continuum answer

  $\displaystyle v({\bf{k}})\Rightarrow{v_{k}}=\frac{H}{{\sqrt{4\epsilon{c_{\rm{s}}}{k^{3}}}}}\left({1+ik{c_{s}}\tau}\right){e^{-ik{c_{s}}\tau}}~{}.$

  (34)

  Note that the existence of the cubic lattice breaks the rotational symmetry, then in the lattice, $v({\bf{k}})$ is not only a function of $k$. We could also compute the next leading short distance corrections of the dispersion relation,

  $\displaystyle\omega({\bf{k}})=k-{b^{2}}\frac{{\sum\limits_{i}{k_{i}^{4}}}}{{24k}}+O({b^{4}})~{},$

  (35)

  which will sequentially modify, for instance, the two-point function and the power spectrum. This is, in fact, the lattice effect of the Lorentz symmetry breaking in cosmic inflation. The lattice effect of quantum gravity is studied in a completely different context. They are assuming the lattice theory should be some ultraviolet completions of general relativity and ask for phenomenological consequences on bounding the lattice spacing $b$, for instance, the observational bound on the power spectrum of CMB Jacobson:2005bg . But now we are completely under a different situation: we are using the lattice theory to simulate the quantum field theory in the continuum.

When we are talking about the flat space, we know that the massless scalar field theory is gapless if we don’t consider the infrared cutoff when defining the continuum field theory. In the lattice version, the gap will scale as $1/L$, which is also the infrared cutoff of the theory. Similarly, a massless field in our inflationary FRW metric is also gapless. Although the Hamiltonian is time-dependent, the Bogoliubov transformation will make the diagonal mode of the Hamiltonian to be still gapless in the whole time during inflation.

A direct way to verify the statement is through a direct computation of the Bogoliubov transformation. To start, we take the free continuum Hamiltonian expressed in $\zeta$ fields and field momenta measured in the $\tau$ coordinates. Replacing the fields and field momenta by their solutions, we obtain the Hamiltonian evaluated in the Heisenberg picture,

where H.c. means the Hermitian conjugate, and

The existence of the off-diagonal term appearing in the Hamiltonian reveals the time-dependent nature of the Hamiltonian. To diagonalize the Hamiltonian, we make the following Bogoliubov transformation

Note that now we keep $a^{\dagger}_{\mathbf{k}}$ and $a_{\mathbf{k}}$ to be static. The coefficient $\alpha$, $\beta$, and the new operators $b^{\dagger}_{\mathbf{k}}$ and $b_{\mathbf{k}}$, are allowed to be time-dependent. We wish to ensure that $b^{\dagger}_{\mathbf{k}}$ and $b_{\mathbf{k}}$ are still canonical

Furthermore, we wish to cancel the off-diagonal terms. We find that we need to demand

Now the Hamiltonian looks like

Ignoring the time-dependence of $b_{\mathbf{k}}$, it seems that the Hamiltonian is formally the same as the flat space. This is because the nature of the above Bogoliubov transformation is to keep re-defining the vacuum state during time evolution evaluated under the definition of $b_{\mathbf{k}}$, to make sure the form of the Hamiltonian is the same as the original time $\tau_{0}$.

The lattice calculation is very similar to the continuum version, since we only need to replace the continuous momenta to the discrete ones. The Hamiltonian is written after the Bogoliubov transformation as

From now, we call the particle created by $b^{\dagger}$ as the “diagonal mode” (normal mode), to distinguish from the particle defined by $a^{\dagger}$, the $\zeta$-particle (the curvature perturbation). The above calculation in the continuum explicitly shows that the Hamiltonian does not have the mass gap at each time slice. Furthermore, from the energy, we cannot distinguish the vacuum and the diagonal modes carrying zero momenta. In the lattice calculation, the gap scales as $1/L$ since our momenta are from the discrete set $\Gamma_{3}$. Those two facts are consistent when we wish to assign an infrared cutoff $1/L$ to the field theory.

We wish to make the following further comments.

• •

  Similar calculations about the above Bogoliubov transformation are done in the circumstance of particle production in inflation, and “cosmic decoherence” by showing similar Bogoliubov transformations, similar to the Unruh effect and black hole information problem, could lead to fruitful particle production. Particles produced by transformations of the vacua are closely related to decoherence where the reduced density matrix becomes nearly diagonal, and quantum perturbations in inflation could become approximately “classical” when exiting the horizon. See some references Gibbons:1977mu ; Abbott:1982hn ; Guth:1985ya ; Albrecht:1992kf ; Lesgourgues:1996jc ; Kiefer:1998qe ; Liu:2016aaf .

• •

  We are performing the calculation in the conformal time $\tau$. The story might be different when we mix time and space and define some other time coordinates. In fact, the definition of the mass gap, which is related to energy, is naturally associated with the definition of time. In general, one might consider defining some versions of the ADM mass, although it might face some technical troubles when we are considering the de Sitter boundary condition. This question might be interesting for quantum simulation since the value of the gap is naturally associated with the efficiency of the adiabatic state preparation process for finding the vacuum. This shows further problems about the compatibility between quantum algorithms using quantum mechanics (requiring a definition of time) and relativity (admitting ambiguities of defining time and space), and the nature of quantum-extended Church-Turing Thesis in the curved spacetime. It might also be interesting to consider different definitions of the gap in AdS. We leave those discussions for future research.

Now we wish to introduce interactions beyond the second-order action. Currently, in the standard theory of inflation, we only assume a scalar field $\phi$, but the nature of this field is not very clear. Furthermore, it might be possible that there are some other particles interacting with inflaton, for instance, the standard model particles. Furthermore, there might be contributions from higher-order cosmic perturbation theory, gravity non-linearities, inflaton self-interaction, and so on. Those contributions will, in principle, give contributions beyond the free theory, and one could those contributions order by order in the perturbation theory framework.

In this paper, we could imagine that we are simulating the following specific model, although most discussions in this paper could be easily generalized to much more generic scenarios. We will consider the general single-field inflation framework, and we expand the action towards the third order Chen:2006nt . We obtain

where $\lambda$ and $\Sigma$ are model parameters⁷⁷7In fact, we have $\displaystyle\Sigma\equiv X{P_{,X}}+2{X^{2}}{P_{,XX}}=\frac{{{H^{2}}\epsilon}}{{c_{s}^{2}}}~{},$ $\displaystyle\lambda\equiv{X^{2}}{P_{,XX}}+\frac{2}{3}{X^{3}}{P_{,XXX}}~{},$ (44) where $P$ is defined in the last footnote. Furthermore, the coupling constants in the cubic interaction are comparable to ${O}(\epsilon^{2})$. . This interaction gives a general form of cubic interaction, which will generate non-trivial three-point functions, namely, non-Gaussianities.

As we discussed in the introduction, the standard inflation theory is expected to be weakly-coupled. Namely, all interactions we include here should have small coupling constants, admitting perturbative expansions. In terms of quantum simulation, during the adiabatic procedure we will discuss later, we could extend the calculation towards the strongly-coupled regime. Thus, we do not really need to expand the action if we really wish to simulate the theory of general single-field inflation: we could even study the original action without perturbative expansion in principle. However, here we are describing an example where we indeed do an expansion before encoding towards quantum devices, with the motivations described in the introduction section of this paper.

Now we write down the Hamiltonian of the above interaction. We have

Up to the leading order perturbation theory, the Legendre transform gives the following total Hamiltonian, split as the addition of the free and the interacting pieces (from now, we ignore the subscript $\tau$ if not necessary)

We wish to split the whole Hamiltonian to three pieces

Here, $H_{1}$ and $H_{2}$ stand for the Hamiltonians represented by purely the field quanta and the field momenta, respectively, and $H_{3}$ is the mixing term. The lattice version is given by

The Hamiltonian $H$ here will be the Hamiltonian we wish to implement in the quantum device. We have the following comments.

• •

  The above splitting $H=H_{1}+H_{2}+H_{3}$ is important when we are doing the Trotter simulation. In the product formula, the Trotter error is estimated by chains of commutators, and the commutation relation about the field and the field momentum will be important to determine those commutators. We will discuss details about this calculation later.

• •

  We owe some higher-order corrections when we are doing the Legendre transformation. That is, the Legendre transformation we have done from the Lagrangian to the Hamiltonian is correct up to the leading order coupling constant. This is because the non-trivial couplings appear in the Legendre transformation. Although we have promised that we are able to simulate the theory non-perturbatively, it should be the Hamiltonian, not the theory defined by the Lagrangian.

Finally, we make some further comments on some aspects of inflationary perturbation theory.

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  The field theory content and its analog in the lattice, described in this section, could be generically applicable for other field theories which are relativistic. One might apply techniques mentioned in this paper to determine problems they are interested in for non-relativistic quantum field theories.

• •

  Choice of momentum/coordinates. When doing quantum simulation, in this paper, we will use the action as the integral over local fields in the spacetime. We write local fields as variables of spacetime coordinates. However, cosmological observables are usually written directly in terms of momenta instead of coordinates. Thus, an extra Fourier transform will be performed classically after we perform quantum simulations of correlation functions measured in different space coordinates. The advantage of such treatment is that we still have a local quantum field theory when we are doing quantum simulation, admit quantum advantage when we are using the product formula to do digital simulations. Of course, one could consider directly simulating the quantum field theory with coupling in the momentum space. However, in this case, we lose our local quantum field theory description. For instance, in this case, we will have a summation of modes of the form $\sigma_{\mathbf{k}}\sigma_{-\mathbf{k}}$, which is not local in the space of momenta. Furthermore, the correlation functions measured in the momentum space will carry a delta function $\delta^{3}(\sum\mathbf{k})$ due to momentum conservation, which is not easy to deal with numerically. In order to read the prefactor in front of the delta function, we probably need to make an extra integral over some momenta variables to integrate the delta function out. This will bring us some extra costs numerically. In any case, to get the full correlation functions, we always need the information specified by different momenta. We leave the direct treatment in the momentum space for future works. For simulating quantum field theories written in the non-local form, see a review in quantum computational chemistry mcardle2020quantum , and recent applications Liu:2020eoa .

• •

  The Bunch-Davies vacuum. Fruitful symmetries in de Sitter space lead to a series of definitions of vacuum states for quantum field theories living in it, which are usually called the $\alpha$-states or the $\alpha$-vacua. Among those states, the Bunch-Davies vacuum plays a special role: it is the vacuum state that looks instantly the Minkovski vacuum, namely, the zero-particle state observed by a free-fall observer. Furthermore, the Bunch-Davies vacuum minimizes the initial energy in our time-dependent Hamiltonian. Thus, the Bunch-Davies vacuum becomes a natural choice when we are doing the cosmic perturbation theory. For cosmological consequences of non-Bunch-Davies vacua, see Easther:2001fi ; Jiang:2016nok for discussions. For recent discussions about simulation of Bunch-Davies vacuum, see Lewis:2019oyx . Exploring non-Bunch-Davies vacua and looking for possible simulation opportunities in quantum technology are important research directions, and in fact, they are closely related to some fundamental assumptions in cosmic perturbation theory, see Jiang:2016nok ; Burgess:2009bs .

• •

  Other coordinate systems. The coordinates we study in this paper are widely used in cosmology literature, but it is only one possible slicing of the global de Sitter. With our coordinate system, it is easy to generalize the flat-space tricks since, in our coordinate, it is manifestly conformally flat. However, one could indeed consider other coordinates. Other slicings and other patches in the de Sitter space might mix the notion of space and time in the current definition. There are similar studies in AdS where people are considering the discretization of the whole spacetime and study their corresponding lattice quantum field theories (see, for instance, Brower:2019kyh , and some previous related research Harlow:2011az ; Gubser:2016guj ; Heydeman:2016ldy ; Bao:2017iye ; Osborne:2017woa ).

• •

  The picture. We are quantizing the field in the Heisenberg picture, and the Hamiltonian is also written in the Heisenberg picture. In the free theory, the mode function satisfies the equation of motion. Then the quantum field satisfies the Heisenberg equation. Note that now the Hamiltonian is completely time-dependent, so the Hamiltonian in the Heisenberg picture is not the same as the Hamiltonian in the Schrödinger picture. In our time-dependent quantum field theory setup, it is more convenient to use the Heisenberg picture, but it is natural to understand quantum computing in the Schrödinger picture. However, it does not affect the current purpose we have. At the first step that we propose in the introduction, we will use the adiabatic state preparation. Since the time is fixed now at $\tau_{0}$, and we slowly turn on the coupling constant dynamically, the Hamiltonian and Schrödinger Hamiltonians are the same. In the second step, we will use the Trotter time evolution to compute the expectation value of correlation functions evaluated at a later time. Then, we could use quantum computation to evaluate the unitary transformation in the Heisenberg picture, acting on the interacting vacuum. Hence, we are mimicking the Heisenberg evolution operator using the Schrödinger evolution of the quantum circuit. It might be interesting to consider using the Schrödinger picture completely in future works.

• •

  Trans-Planckian problem. How early should we set $\tau_{0}$ when we are doing the simulation? We know that the comoving scale should be exponentially small in the past since the universe is exponentially expanding. Thus, early time should correspond to extremely short distances, which might be smaller than the Planck length. This is called the Trans-Planckian problem in cosmology, since we don’t have a Planckian theory of quantum gravity Martin:2000xs . We are not addressing the conceptual resolution to the problem, but it is practically related to where we could choose $\tau_{0}$ when we are doing the simulation. In fact, this could be understood as an advantage of our simulation. We expect $\tau_{0}$ would happen at the very far past, and we could test different values of initial time $\tau_{0}$ other than the ideal $-\infty$ when we are doing the perturbative quantum field theory. This might be helpful for theoretical physicists to address the Trans-Planckian issue. For instance, one could check if the vacuum is really adiabatic on sub-horizon scales. In fact, this issue is closely related to the in-in formalism we will discuss later since we care about interactions in the Trans-Planckian regime happening at an extremely early time⁸⁸8The adiabaticity of the vacuum here is not exactly the same as the adiabatic quantum computation we will discuss later. In the later treatment, we artificially turn on the interactions, so it is a time-dependent dynamical process at the fixed initial time. But the adiabatic vacuum is made by the time-dependent quantum field theory we discuss here introduced by the scale factor..

The in-in formalism is a powerful approach for computing correlation functions for the time-dependent Hamiltonian, and it is specifically useful for cosmic perturbation theory. Unlike the S-matrix in the flat space, which starts from infinite past to infinite future (the in-out formalism), the in-in formalism is concerning the correlation functions where bras and kets are both from the past, namely, both the “in states”.

As we know, our observational data about the universe should, in principle, come from observables in quantum mechanics. Generically, we wish to compute the following expectation value in the Heisenberg picture,

where the operator is now evolving with time, and the state is static, equivalent to the state at the initial time $\tau_{0}$.

In this section, we will describe how to evaluate the above variable in the cosmic perturbation theory, how it is connected to observations, and how it is related to quantum simulation. Since our theory could generically apply to time-dependent systems, we will use the notation $\tau$ for time, and when applying the theory to cosmic inflation, we will use the in-in formalism completely in the conformal time.