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Quantum Simulation of Conformal Field Theory

Conformal field theory, describing systems with scaling symmetry, plays a crucial role throughout physics. We describe a quantum algorithm to simulate the dynamics of conformal field theories, including the action of local conformal transformations. A full analysis of the approximation errors suggests near-term applicability of our algorithm: promising results for conformal field theories with central charge c=1/2 are obtained already with 128 logical qubits.

for the energy scale -the renormalization scale -set by the lattice spacing. The extraordinary insight of Wilson is that consistency under changes of scale gives the definition of an ultraviolet completion -a procedure known as the renormalization group. This powerful strategy has been employed throughout computational quantum field theory, with many dramatic results, especially for massive QFTs (18,19). However, effective field theory doesn't provide us with all the answers. In particular, for theories with scaling symmetries mixing the infrared and ultraviolet it is necessary to understand the continuum limit.
Conformal field theory (CFT) (20,21) epitomises the outstanding challenges facing the development of quantum simulation algorithms (14,22). In particular CFTs are massless and, in 1 + 1 dimensions, have an infinite-dimensional group of local conformal symmetries. Thus the simulation of their dynamics and continuous scaling symmetries via discretised approximations is particularly problematic as one needs to understand or construct the continuum limit to assess the accuracy of approximation. Fortunately, we are on the cusp of a satisfactory mathematical formulation of CFT, providing us with a unique opportunity to compare discrete approximations directly with rigourous continuum results (22)(23)(24)(25)(26)(27)(28). Further, there is a rich supply of physically relevant examples to prototype quantum simulation algorithms, with a vast range of applications from phase transitions to quantum gravity (29,30).
We meet the above challenges (14,22) and introduce a new aproach to the quantum simulation of the dynamics of CFTs. We show how to overcome, by appealing to the recently developed operator-algebraic renormalization (OAR) (27,31,32), the twin difficulties presented by the comparison with the continuum limit and the implementation of local scaling transformations within a lattice formulation. We introduce a variety of techniques to rigourously construct the continuum limit and provide a quantum algorithm to coherently simulate the action of local conformal transformations on the many body Hilbert space of CFT. Remarkably, we find that the resulting quantum algorithm can provide promising approximations on near-term quantum hardware with as few as N = 128 logical qubits (see Supplementary Material).
The additional symmetries provided by the conformal group delivers a powerful calculational tool for the computation of physically relevant correlation functions. Indeed, two-and three-point correlation functions are all but completely determined (20,21). However, the computation of higher correlation functions becomes rapidly expensive (33). By simulating the entire physically relevant Hilbert space at a given spatial scale we are not only able to efficiently calculate four-point and higher correlators, but also more complicated observables. The complexity of the quantum algorithms we present here scales as δ −1 with the accuracy δ required 1 .
The crucial device exploited in this paper is OAR, which is a mathematically rigourous formulation of Wilsonian renormalization in the Heisenberg picture of observables, where locality is manifest. The core innovation of OAR is that the quantum continuum limit is constructed as a sequence of discrete approximations, directly supplying us with error estimates. This powerful formalism is also extremely flexible, allowing for formulations, both in momentum space and -via approximations built using wavelets (34) -position space. Here we generalise OAR to implement discretised approximations for local conformal transformations via a remarkable formula due to Koo and Saleur (35).
The quantum algorithm presented here may be compared with efficient classical algorithms based on tensor networks. Very recent progress has led to dramatic progress in the classical simulation of CFTs, in particular, for the computation of equal-time vacuum correlation functions, including both rigourous approximations and numerical procedures (36)(37)(38)(39)(40)(41)(42)(43). However, a core obstruction in the classical setting remains the simulation of quantum dynamics and conformal symmetries: these processes easily create massive entanglement, obstructing their simulation via tensor networks which scale, in the worst case, exponentially with the number of lattice sites.
In this paper we exploit OAR to formulate a quantum simulation algorithm for conformal field theories in 1 + 1 dimensions to compute approximations to quantum field observables, local either in momentum or position space. Further, we are able to simulate correlation functions for observables which are unbounded in the continuum, allowing us to consider correlation functions for the Virasoro algebra. A key novelty of the approach described here is that unitary local conformal transformations may be directly simulated and errors estimated. We focus on a fermionic example with central charge c = 1 2 , however, we also explain how to extend our results to cover currents for Wess-Zumino-Witten models. Our approach admits natural generalisations to a wider class of models with arbitrary central charge, including bosonic models and models based on anyonic systems.
Representing conformal fields on a quantum computer. We exploit the Hamiltonian formulation of conformal field theory and work with units so that = c = 1 (21). In order to simplify the exposition we illustrate our approach using the key example of relativistic Dirac fermions on the circle with circumference 2L. Depending on the boundary conditions -i.e., either periodic or antiperiodic -one realises the Ramond (respectively, Neveu-Schwarz) sector. The real chiral components of the massless theory give rise to CFTs with central charge To simulate a CFT on a quantum computer we need to approximate the system with a finite number of qubits. This is achieved by discretising the system onto a finite spatial lattice Λ N = Exploiting the staggered lattice approximation we may discretise the theory with the following lattice hamiltonian: where λ N is the lattice mass coupling constant (the prefactor has been chosen for consistency with the Koo-Saleur formula). Note that N refers to the spatial scale, i.e., the negative logarithm of the lattice spacing. The total number n of lattice sites at scale N is given by n = 2 N +1 . The lattice fermion operators ψ (j) x , j = 1, 2, obey the standard canonical anticommutation relations: Here the fermions are defined with respect to the lattice Λ N +1 with half the original lattice spacing. In this way the Hamiltonian may be simulated via the following quantum spin system (48) The term "B.C.s" refers to the nontrivial term − 1 encoding the possible boundary conditions (with the first sign referring to spins and the second to fermions). It is noteworthy that anti-periodic fermions complemented by periodic spins lead to (1) a vanishing boundary term on states with even fermion parity, specifically the Fock vacuum, and (2) a non-degenerate ground state in the massless case (Neveu-Schwarz sector). We note that there is considerable room for optimisation here, and one can expect that more sophisticated representations will deliver improved resource requirements (49).
Description of the quantum simulation algorithm. After choosing a desired accuracy δ for the result of the simulation the algorithm consists of the following four steps, which we describe and analyse in the sequel.
1. Prepare the initial state of the lattice. There is a wide class of possibilities, including the ground state, which can be directly achieved using an efficient quantum circuit.
3. Evolve the system according to a discretised local conformal symmetry transformation, via the Suzuki-Trotter decomposition.
4. Exploit phase estimation to measure local field operators, either in real or momentum space.
Complexity. The computational complexity, i.e., the asymptotic scaling of the number of quantum gates required to compute a correlation function, of the algorithm we present here depends on several parameters. The most important is the desired accuracy δ of the simulation, which measures the absolute difference between the computed discretised approximate results and the theoretical target value where |Ω is the CFT state (which need not be the vacuum), A and B are field observables, local in either real or momentum space, the subscript t denotes that the observable has been evolved in the Heisenberg picture according to the dynamics generated by a local conformal transformation, e.g., B t ≡ e itL k Be −itL k , where L k is a Virasoro generator, the tildes represent discretised approximations, and N = log 2 (n) − 1 represents the number of lattice sites of the discretised approximation. By increasing the number n of lattice sites one can reduce the errors according to 1/n 2 : this provides the direct connection between the complexity of the simulation and the accuracy. There is an implied constant here which depends on the order k of the local conformal transformation desired. One can generalise these error bounds to cover multipoint correlation functions with m operators; a simple bound shows that the errors grow quadratically with m. An important feature of our approach is that A and B are not required to be bounded operators in the continuum limit: our results also apply to the case where, e.g., A and B are unbounded Virasoro generators.
It is important to note that there is no mass gap determining a length scale for CFTs. Since ∞ on A N , which is stable under coarse-graining 3 : Note that finding a limit state usually requires that the coupling constants of the bare model be redefined at each scale according to the renormalization conditions. We are ultimately interested in the double scaling limit ρ ∞ , which realises the continuum limit (this exists as soon as the scaling limit exists because of (3), so this does not entail additional constraints on the construction). The scaling limit of the family H (N ) 0 of lattice Hamiltonians is naturally understood via a limit, τ (∞) t , in the Heisenberg picture of the discretised approximations τ To exploit OAR we need to select a family of coarse-graining/scaling maps E which are compatible with the physics we expect in the continuum scaling limit; the success of OAR requires that one retains enough of the quantum coherence present in low-energy degrees of x | x ∈ Λ N , j = 1, 2 . We then write the RG maps for our real-space scheme according to where {c l } l∈Z is the low-pass filter determined by a scaling function s associated with (compactly supported) orthogonal wavelets 4 (34) An explicit example is furnished by the Daubechies D4 wavelet, for which the coefficients are However, in order to simulate observables which, in the continuum limit, are unbounded, it can be necessary to choose wavelets with higher regularity than D4. Note that the renormalized Figure 2: Operator algebraic renormalization in real space for the wavelet D4. Here ε labels the lattice spacing of the coarse lattice, α N N +1 and E N +1 N denote the RG maps in the Heisenberg (respectively, Schrödinger) picture, and c j are the low-pass filter coefficients defining the D4 wavelet.
x ) acts on the fine-grained lattice, with lattice spacing 1 2 ε. The resulting real-space OAR is illustrated in Fig. 2.
To describe the momentum-space RG maps we first introduce the Fourier transformed op- according to which the RG acts via where χ M (k) = 1 if k ∈ π L {−L M + 1 2 , . . . , L M + 1 2 } and 0 otherwise. That is, only operators ψ Discretised Virasoro generators. With the basic kinematical and dynamical data in hand we turn now to the quantum simulation of conformal symmetries. These form the conformal group with the associated Virasoro algebra of generators spanned by L k , k ∈ π L Z, satisfying where c is the central charge. Several of the generators are directly related to more familiar observables, for example, the CFT Hamiltonian and total momentum are given by To approximate L k on a lattice at scale N we exploit an insightful ansatz due to Koo and Saleur (35); we begin by introducing the fourier modes of the lattice Hamiltonian density 5 , which is symmetric around x, to obtain the correct Virasoro algebra in the scaling limit: and then defining with an analogous formula for L  (48). 5 In the case of massless Dirac fermions (λ N = 0): h The crucial observation underlying our quantum simulation algorithm is that L (N ) k so defined in (11) are sums of densities. This means that we may employ standard quantum algorithms for the simulation of the dynamics of local hamiltonians to simulate the exponentials of these generators 6 ; since the behaviour of these methods are well understood we focus on the new challenges arising in approximating the continuum limit with a lattice model.
Preparing the initial state. The first step is to initialise the 2n qubits of the quantum computer into a discrete approximation of the desired state. We focus, for concreteness, on the approximation |Ω N of the ground state |Ω of the scaling limit, and states in the conformal block found by applying Virasoro generators to |Ω . (It is important to note that our algorithm is by no means limited to such states, and can be applied to any quantum state with a scaling continuum limit.) There are now several approaches to the efficient preparation of |Ω N . A particularly parsimonious quantum circuit, which we employ here, is adapted from (51) (see Fig. 3): Note that the quantum spin Hamiltonian (2) may be diagonalized by a quantum circuit for a (fast) fourier transform U FT followed by a Bogoliubov transformation U B , so H is diagonal. In this way we may realise the ground state from a product state: With the qubit registers in the discrete approximation of the ground state we now apply the desired generators A and B for the correlation function to state |Ω N . For concreteness we focus on the momentum-space RG. In the case of correlation functions for operators ψ 6 In order to ensure that the resulting dynamics are unitary we actually simulate hermitian linear combinations of L FT to the momentum-space representation of the ground state U FT U B k∈Γ N +1 | ← , and then measuring the σ x operator on the ancilla qubit, postselecting on the positive outcome. Repeating this procedure a constant number of times will ensure the correct state is produced with high probability (for a constant number of field operators). In this way we may easily prepare the initial state k l ψ (j l ) k l |Ω N . (The application of a field operator local in real space is more involved and requires consideration of the Wavelet basis employed in the OAR. See (48) for further details.) The resulting initial state is denoted |Φ init .
Simulating dynamics. Now that we have prepared the initial state we may apply local conformal transformations. Formulating the Fourier-transformed fermion operators in terms of the single-component fermion, ψ ), these are generated by hermitian linear combinations of the following Koo-Saleur operators (N ) These generators are (2k +1)-local in the momentum representation and may be simulated via a Suzuki-Trotter decomposition. (A state-of-the-art implementation will make use of the optimal simulation algorithms.) After application of the generator the quantum register is in the state Note that the k = 0 case reduces to simulating the unitary Hamiltonian evolution.
Measuring observables. To measure a desired observable O, e.g., O = ψ (j) k , one may employ standard phase estimation (52,53). This requires the addition of r ancilla qubits to represent the readout, where r is the required to be large enough to resolve all of the eigenvalues of O.
Repeating this procedure delivers an estimate for the expectation value k ) via, e.g., product formulas (54), and may be reduced by exploiting higher-order approximations. Exploiting state-of-the art estimates, one requires quantum gates (nt) 1+o (1) to reduce this error to below δ/2. The state preparation is considered exact (up to the gate compilation required to implement the single-and two-qubit unitaries required). The fourth source of error arises from the discretization itself. This is deeply nontrivial and requires an extensive and challenging analysis, discussed in the companion paper (48). The result of this analysis implies the existence, for a given desired accuracy δ and t ≤ T , of a constant size N (δ, T ) lattice achieving this accuracy. The dependence of N on δ and T is nontrivial, however, numerical experiments -see the Supplementary Material -indicate that, e.g., a D10 wavelet can achieve an accuracy of δ = 1/2 on a lattice of 128 sites. (This is for the case where one computes dynamical correlation functions involving Virasoro generators.) Extensions. One may generalise the presented algorithm to approximate the lattice currents of Wess-Zumino-Witten models. For example, to target the level-1 WZW model for u(D) 1 , we require D-component fermion fields, which we can simulate using D copies of the realisation exploited above. Details of this approach may be found in (48). The Ising CFT may be directly targeted using a modification of the approach here. The details of this will be presented in a follow up publication. More generally, to simulate an arbtirary rational CFT one must take recourse to models involving chains of anyons. To employ OAR we need an appropriate family of coarse-graining maps. Such a family is induced by the Jones-Wenzl projection, however, this is beyond the scope of this paper and will be presented elsewhere.   on one-particle states corresponding to the subspace of the CFT Hilbert space representing momenta up to the simulation scale M as a function of the UV cutoff scale N . Note, that this error vanishes for the generators of the Moebius group, k = 0, ±1. For further details, see (48).