Quantum algorithms for conformal bootstrap

With the help of recent developments in quantum algorithms for semidefinite programming, we discuss the possibility for quantum speedup for the numerical conformal bootstrap in conformal field theory. We show that quantum algorithms may have significant improvement in the computational performance for several numerical bootstrap problems.


Introduction
Quantum computing is a rapidly growing area both theoretically and experimentally. In the near future, people will be able to manipulate O(50) ∼ O(100) qubits, a regime which could potentially run algorithms that can outperform classical computers, even with quite high levels of noise. The coming era, the so called Noisy Intermediate-Scale Quantum(NISQ) era, may therefore allow physicists to simulate quantum physics and solve physics problems too difficult for classical computers [2].
One of the earliest motivation for designing quantum computers was the simulation of quantum field theory [1]. With infinite degrees of freedom and complicated mathematical structures, this is a problem that has traditionally been difficult for classical computing. Some interesting work has been pursued as to whether quantum algorithms, by contrast, could be designed to simulate field theory processes, such as time evolution of wave packets, and scattering problems in high energy theory (for example, see [3][4][5][6][7]). Indeed, foundational aspects of quantum algorithmic design (such as complexity theory) could correspond to deep concepts in quantum field theory, holography or quantum gravity (for example, see [8][9][10]).
For special classes of quantum field theories, such as theories with conformal symmetry, non-perturbative technologies are widely used, that sometimes allow for a solution to the theory through either analytical or numerical techniques. A leading example of this prevalent in the current high energy theory community is called the conformal bootstrap. The conformal bootstrap is an approach that extracts physical data, such as dimensions of states and Operator Product Expansion (OPE) coefficients, from internal non-trivial consistency relations called crossing equations derived from conformal symmetry. Currently, the conformal bootstrap is widely used in conformal field theory (CFT), string theory and statistical physics (for example, see [11][12][13][14][15][16][17][18]). One of the most important developments in this field has been the advent of the numerical bootstrap as a method for finding high precision numerical solutions for critical components in the three dimensional Ising model, an old but interesting puzzle in statistical physics [11].
For completeness, we review the basic form of the conformal bootstrap problem. In its simplest case, the crossing equation, derived from the crossing symmetry of the conformal four point function, can be written in the following form where the x's are particular combinations of spacetime points, F is a known function, and the summation over ∆ and J (corresponding to the spectrum of the operator appearing in the OPE) are not known. Alternatively, we could also have known operators in the spectrum. Typically, however, we only know the identity operator, ∆ 0 and J 0 .
The goal is then to solve the spectrum from such a crossing equation. In general, the function F is complicated enough such that determining the spectrum analytically is intractible. Despite this, however, numerical methods have proven successful for this problem. The particular method used here is based on semidefinite programming, a widely used method for operational research and related areas. We can begin by writing a generic form of the linear functional and applying this functional to each side of the crossing equation. If for ∆ > ∆ * , one can always find a linear functional such that then a contradiction is reached. This would therefore mean that there must exist a state with dimension ∆ < ∆ * . Thus, we could search for the minimal value of ∆ * , which corresponds to the gap of the theory. The functional searching with positivity condition, can be transformed to a standard form of semidefinite programming [13].
In this paper, we will address the possibility for quantum speedup for the numerical conformal bootstrap. This is motivated for several reasons. First, some bootstrap problems are extremely difficult to solve (for instance, [19]) on classical computers, and it is possible that quantum computing techniques are better adapted to these questions. Second, solutions of certain numberical bootstrap problems may require higher precision in order to compare against Monte Carlo results and experiments for statistical models. Recently, rapid developments have shown that semidefinite programming indeed has the potential for dramatic quantum speedup [21][22][23][24]. We will make use of those algorithms to address specific problems in the numerical conformal bootstrap.
This paper is organized as follows. In Section 2 we will review the conformal bootstrap, the Positive Matrix Program(PMP), and its connection to semidefinite programming(SDP). In Section 3 we will discuss the fasted current quantum algorithms for solving SDPs. In Section 4 we will study the particular speedups in the context of numerical bootstrap problems by quantum algorithms. In Section 5 we will give an explicit example, the relatively well-studied three dimensional Ising bootstrap, to discuss in more specificity the potential for quantum speedup. Finally, in Section 6, we conclude and discuss several future research directions.

Conformal bootstrap and semidefinite programming
In this section, we will review some basics about the conformal bootstrap program and semidefinite programming from [13]. The conformal bootstrap problem can typically be written in the following form. Consider a given symmetric polynomial matrix set where n ∈ {0, 1, · · · , N } and j ∈ {1, · · · , J}, and the P n j,rs (x) are all polynomial. Thus, given b ∈ R N , one can run the following Positive Matrix Program(PMP): Where the symbol M 0 means that M is positive semi-definite.
Here we can give an example about how the conformal bootstrap can be transformed into the PMP. Consider the 3D Ising CFT. In this theory, the crossing equation for σσσσ , σσ and will be given by Note here that O + runs over the Z 2 even operators with even spin, while O − runs over the Z 2 odd operators with all possible spins. Further, ∆, are the dimension and spin of the operator O. The vectors V is then defined with the following form while the function F is given by a combination of g, the conformal blocks, which are known functions: where we denote ∆ ij = ∆ i − ∆ j . Now consider a functional vector α = (α i ), for i = {1, 2, · · · , 5}. We now act α on the crossing equation Then, if we could find an α such that then the theory is ruled out; this method can therefore be used to rule out some range of ∆'s from being in the allowed theory space.
In order to transform this problem to a PMP, we will consider the following linear functional where u = zz and v = (1−z)(1−z). Note that derivatives of function F are well approximated by polynomials, with positive prefactors where P s are polynomials and χs are positive prefactors. Thus we arrive at a standard form of the PMP, 3DIsing: find: a i mn such that: Note that ∆ could be shifted to some minimal value ∆ min ( ) plus x, and that we can demand x ≥ 0 to replace the requirement that ∆ ≥ ∆ min ( ).
There are several thing to note in this example. First, we only care about feasibility, and therefore we are allowed to set the objective function to zero in PMP. Other bootstrap problems, for instance the finding of bounds over OPE coefficients, could in general need a nontrivial objective function. Second, the set of spin is infinitely large, and so we should truncate it to be a finite set, for instance ranging from 0 towards max . For the bootstrap, if max is sufficiently large, the positivity for all spins is typically ensured.
Finally, we will review how to relate PMP's to SDP's. Here we will use SDP's of the following form: for given c ∈ R P , B ∈ R P ×N , and A 1 , A 2 , · · · , A P , C ∈ S K . Here S K is the space of the real K × K symmetric matrix, and we write A * = (A 1 , A 2 , · · · A P ). To transform our PMP problem into this form, we will eventually set C = 0, although we will allow for it to remain general for now.
We quote the theorem for transforming PMP to SDP reviewed and discussed in [13]: Given a PMP problem, if one sets p to be tuples (i, j, k, l) with 0 ≤ r ≤ s < m j , 0 ≤ k ≤ d j and 1 ≤ j ≤ J, then this tranforms to an SDP of the form , is a collection of polynomials with degree 0, 1, · · · , δ.
Based on the theorem, we know that the volume of matrices, as input, are given by where we assume that M is m × m matrix with maximal degree d, and N in the PMP is the same as that of the SDP.
There are several polynomial time classical algorithms for solving SDP problems. In the current conformal bootstrap community, people often use SDPB, an efficient PMP solver proposed in [13] to solve bootstrap problems based on the above transformation from PMP to SDP. The SDP solver in [13] is based on the exact interior point method. The complexity of the algorithm, within a reasonably small error, is known to be O(

Quantum algorithms
Recently rapid progress has been made in finding a quantum version of an SDP solver. The first quantum SDP solver was giving in [21], and further developments refinements and speedups were found in [23]. In addition, an independent group obtain related results with different formulations and input models [22,24]. In this section, we will describe two quantum algorithms most following [23] for solving SDP that could be useful for the numerical conformal bootstrap, while an improvement with a stronger input model in [24] is also mentioned.
When constructing quantum algorithms, we first need to specify how to input the elements of the matrices and vectors for a given SDP into the quantum computer, namely, how to compile the information of matrix and vectors into the quantum circuits. Those are called Input models or Oracles. The first algorithm we will describe uses an oracle called the plain model, while in the second algorithm uses a quantum input model. Those two oracles, perhaps unsurprisingly, lead to algorithms with different complexities. The quantum input model is a stronger oracle that applies in a more restricted set of cases, and thus the corresponding algorithm is faster in its scaling with certain parameters. If we have a good quantum computer, which oracle one chooses will sensitively depend on the detailed quantum architecture at hand.
Before going into detailed discussion, let us first remark on two issues. First, in the current literature for quantum SDP solvers, people use the language of complex(Hermitian) matrices instead of real(symmetric) matrices. However, with some simple modifications (for instance, replacing the Hermitian matrix with a real symmetric matrix in the proof of theorems, or using a real Gibbs sampler instead of a complex one, etc.), one can easily adapt those algorithms without changing the complexity analysis.
Second, people use different forms to describe equivalent SDP problems. In the following descriptions, the first two algorithms are written in terms of SDP with the form of the Semidefinite Programming (Approximate) Feasibility Problem (SDPFP) SDPFP: Find Tr(ÂîX) ≤âî + for allî ∈ {1, 2, · · · ,m} such thatX 0 Tr(X) = 1 for given > 0 ,Âî ∈ Sn , −I Âî I (3.1) Here, we use the hat notation to distinguish from the variables we used above. The above conditions define a convex region S for given . The result of the algorithm will be: (1) If S 0 = ∅ we will output Fail; else, (2) if S 0 = ∅, output anX. In the above notation,m is the number of constraints,n is the number of dimensions, and denotes the additive error of a solution. Further, we use A B to denote the condition that A − B is positive semidefinite, while similarly A B denotes that B −A is positive semidefinite. I is the identity matrix. We choose the dimension of matrixn, number of constraintm, and the number sparsity (namely, take the maximal number of non-zero elements in each row of each matrix) ofÂ i to beŝ.
Several versions of SDP problems are equivalent and simply related. We discuss them in Appendix A.
Trace estimator: The input of the trace estimator is a real symmetric matrix H, satisfying ||H|| ≤ Γ, and a density matrix ρ. Then one can compute Tr(Hρ), with additive error , and success rate at least 2/3. We write S Tr (ŝ, Γ, ) to denote the sample complexity.
Gibbs sampler: The Gibbs sampler takes as input a a real symmetric matrix H, satisfying ||H|| ≤ Γ and prepares as output a Gibbs state ρ = e −H Tr(e −H ) with additive error with the help of plain model oracle.
Now based on those building blocks, we will briefly describe the algorithm constructed in [23].
Algorithm 3.2. Quantum SDP solver with plain model: • Choose an initial weight matrix W (1) = In, and choose T = 16n 2 .
• Start a loop, that for t = 1, t ≤ T , t = t + 1: -Apply the Gibbs sampler O(logm×S Tr (ŝ logn 2 , logn , )) times, and get several ρ (t) = Tr(W (t) ) . -Use these copies of ρ (t) to find aĵ (t) such that Tr(Âĵ (t) ρ (t) ) >âĵ (t) + , where we use the trace estimator to compute the trace. If we find suchĵ (t) , we update M (t) = 1 2 (I n −Âĵ (t) ). If we cannot find it, then we say that S is not empty and output ρ (t) asX, the solution.
• We say that S is empty.
We note that there is an important procedure for finding a suitableĵ (t) . This procedure can be implemented by a fast quantum OR lemma. For details, see [23].
The gate complexity we need for running this algorithm is given by O(ŝ 2 ( √m 10 + √n 12 )), if we use the real version of the Gibbs sampler from [25] and the trace estimator from [21].
In [24], the corresponding algorithm is similar, but they use a different but stronger input model, the quantum operator model, defined by the following.
Oracle (quantum operator model): For matricesÂĵ, find an operator U such thatÂĵ = α( 0| ⊗a ⊗ I)Uĵ(|0 ⊗a ⊗ I) for some given α and a. The oracle O U is then defined such that With this input model, the runtime required is reduced to O(ŝ 2 ( √m 4 + √n 5 )).

Quantum SDP solver with quantum input model
Now let us consider another quantum solver, this time with a quantum input model. Taking advantage of this input model, the gate complexity will be reduced for some input parameters.
We assume thatÂĵ can be decomposed asÂĵ =Â + j −Â − j , whileÂ ± j 0.This assumption will always hold whenÂĵ is a real symmetric matrix. In fact, one can take ζ large enough such thatÂĵ + ζI is a diagonally dominant matrix. Since it is diagonally dominant and it is symmetric, it is positive semidefinite. Thn we can simply takeÂĵ + ζI =Â + j and ζI =Â − j . In our problem, we know the input is bounded by A naive choice is to set ζ = 1; better decompositions may exist, however, for specific problems.
We will now describe the building blocks of the algorithm.
Oracle (quantum input model forÂĵ): We define a quantum oracle O such that Oracle (quantum input model forâĵ): We define a quantum oracle Oâ such that We assume that in the input we haveÂĵ has rank at mostr and Tr(Â + j ) + Tr(Â − j ) ≤B where B can be understood as the upper bound of the norm for all inputs. In our problem we havê Moreover, we define Trace estimator: The input of the trace estimator is a state ρ, and the output will give, within success probability 1 − O(1/m), whether Tr(Âĵρ) >âĵ + or Tr(Âĵρ) <âĵ. We denote S Tr (B, ) as the sample complexity. The estimator is defined based on the quantum input models defined above.
Based on those building blocks we have Algorithm 3.3. Quantum SDP solver with quantum input model: • Choose an initial weight matrix W (1) = In, and choose T = 16n 2 .
• Start a loop, that for t = 1, t ≤ T , t = t + 1: -Apply the Gibbs sampler O(S Tr (B, )) times, and get several ρ (t) = W (t) Tr(W (t) ) . -Use these copies of ρ (t) to find aĵ (t) such that Tr(Âĵ (t) ρ (t) ) >âĵ (t) + , where we use the trace estimator to compute the trace. If we find such anĵ (t) , we then update M (t) = 1 2 (I n −Âĵ (t) ). If we cannot find it, then we say that S is not empty and output ρ (t) asX, the solution.
• We say that S is empty.
This algorithm is very efficient based on the Gibbs sampler and the trace estimator that are constructed in [23]. Its complexity is given by A remarkable fact here is that it only has the polylog dependence onm andn regardless of B.

Specifying bootstrap problems
When discussing the bootstrap problem, we can start from PMP problems. Similarly, the quantum algorithm with plain input model works for the SDP problem, and thus works for PMP problems. The runtime estimate here would be Thus, generically the quantum algorithm with plain input model will provide a quantum speedup.
For the quantum input model, we know that the rank of the inputÂĵ is at most, So it is always faster than the two given classical algorithms. For large J, it is slower than the plain input model, but it is faster for large d, m, N (holding all other parameters fixed).
• In the worst case we considerr = O(d + N ) andB = O(Jmd + N ). We denote the degree of polynomial dependence onr andB to be d r and d B separately. Thus, whether the algorithm is fast or not, is based on the following conditions Now we take a look at the bootstrap problem itself. A typical bootstrap problem will have the parameter set where Λ is the cutoff of the number derivative. It scales as a square because we have both z andz. Thus we have complexities We should comment briefly on those results. First, we see that requiring a quantum speed up and advantage over the plain input model strongly constrains the allowed quantum oracle algorithms. The reason is that for bootstrap problems, we typically require the input matrix to have a large rank, which effectively negates the exponential advantage of the quantum oracle algorithm. On the other hand, bootstrap problems often have lowŝ (namely, the input is very sparse), because the sparsityŝ has a very weak dependence on the large numbers (linear in d). That means that the square root speedup of the plain input model algorithm will be generically robust for such problems.

Examples
It is valuable to know how parameters scale in the actual bootstrap problems. We will provide some examples here.
Crossing of σσσσ in 3D Ising: Here we consider only the first crossing equation in the 3DIsing problem. In this special case, positivity of matrices become positivity of numbers. Namely, the problem now becomes In this problem, we will label the maximal number of derivatives to be nmax, while the maximal spin we take to be lmax (The relation between nmax and Λ is given by Λ = 2 × nmax − 1). For simplicity, we consider lmax to be even and take the parameters to not be too small. The parameters in the theory will then scale as the following, The results agree with out expectation of the generic analysis in Section 4. We plot the SDPFP parameterm,n,ŝ,r in Figure 1.
The full 3D Ising problem: A more serious analysis can be done for the full 3DIsing problem. Note that here in the crossing equation we have to consider the possibility that O + is equal to σ and O − is equal to in the sum. As a result we obtain a similar table, Parameters Values Similarly, we plot them,n,ŝ,r parameter in Figure 2. The value of N matches the dimension of α coefficients around equation (3.15) of [13]. Moreover, we could use the parameter setup used in Figure 1 of [13], which is given in the following table. As we mentioned before, we see that the sparsity growth is relatively slow (linear in nmax with low slope). This helps to retain the advantage of the plain input model, since it depends on sparsity byŝ 2 , but it depends onm andn with square roots.

Conclusion and outlook
In this paper, we address the possibility of using quantum algorithms for SDP problems to numerically solve crossing equations for the conformal bootstrap. Generically, we show that quantum algorithms will provide significant speedup over classical SDP algorithms for specific conformal bootstrap setups. We expect that these methods may have possible applications in the coming NISQ era, and beyond, to help the bootstrap community obtain more accurate and faster numerical implementations.
The area of quantum SDP is still young. In this paper, we mainly discuss two quantum algo-rithms. However, in the near future, it is reasonable to believe that more efficient algorithms and more realistic constructions with corresponding quantum architectures will be developed.
We will leave the following possible directions to future research.
• Oracles. It will be important to construct oracles (unitaries) to suppy the bootstrap inputs to the quantum computers, and the complexities of implementing these oracles will also constraint the overall algorithmic complexity at hand.
• Upgrading the Gibbs sampler and the trace estimator. One could potentially develop better methods for speeding up the Gibbs sampling and trace estimation, or think about how to realize them in actual quantum devices.
• More specified solver. The bootstrap problems take a special shape of PMP, which is a special category of SDP. It is therefore hypothetically possible to construct algorithms that taken advantage of this specific from to further speed up the computation.
• Realizations. It might be useful to perform the above algorithms in some quantum simulators, or real quantum devices in the future, to empyrically test whether the theoretical predicitions for runtimes are perhaps too conservative. An attempt to adapt classical SDP solvers for the conformal bootstrap is already underway in [20].
where A P (1) ). Firstly we show that it could be transformed as SDP2: maximize: Tr(C (2) Y (2) ) on Y (2) ∈ S K (2) such that: Tr(A In fact we write y i − s (1) , where z (1) and s (1) are positive. Then we have ? and Tr(A ?
where ? means it is not necessary to know. So we only need to set Now one could also show that it could be converted to such that: Tr(A In fact, if we have a solver for SDP3, let us take A (3) to be (A i ) (Note that now the number of A (3) is doubled), and we set c (3) to be (c (2) , −c (2) ). That means that we impose the constraint So it gives the equality.
These operations mean that we could directly use the solver SDP3 to solve the problem of SDP1. We need to just set Finally we will show that we could solve SDP1 using the SDPFP solver, SDPFP: Find Tr(ÂîX) ≤âî for allî ∈ {1, 2, · · · ,m} such thatX 0 Tr(X) = 1 for givenÂî ∈ Sn , −I Âî I (A.11) We could start from SDP3. We notice that for given A (3) there exists a positive number λ such that −λI A λ and we should also redefine c (3) by c (3) λ . For optimization, we could using binary search by assuming for instance, Tr(C (3) Y (3) ) ≥ q bin or namely, Tr(−C (3) Y (3) ) ≤ −q bin and treat this as a component of A (3) * . We could increase q bin until we cannot find the optimal solution.
Finally, for the condition Tr(X) = 1, we could treat it as the following. Firstly, the solver for Tr(X) = 1 means that we have a solver for Tr(X) =ω whenω is an arbitrary positive number, since we could redefineâî byâˆî ω . For a given SDP3 problem, it does not matter if we add a constraint Tr(Y (3) ) ≤ω ifω is sufficiently large. Then we could use the slack variable trick by adding variable µ ≥ 0 and transforming the inequality to the equality Tr(Y (3) ) + µ =ω.
As a conclusion, for given SDP1 problem, we define and then we find a large positive λ such that −λI A i , C (3) λI (A.14) for all i. Then we set