Learning optimal quantum models is NP-hard The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Stark, Cyril J. et al. "Learning optimal quantum models is NP- hard. " Physical Review A 97, 2 (February 2018): 020103(R) © 2018 American Physical Society As Published http://dx.doi.org/10.1103/PhysRevA.97.020103 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/114464 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW A 97, 020103(R) (2018) Rapid Communications Learning optimal quantum models is NP-hard Cyril J. Stark Center for Theoretical Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307, USA (Received 19 August 2016; published 8 February 2018) Physical modeling translates measured data into a physical model. Physical modeling is a major objective in physics and is generally regarded as a creative process. How good are computers at solving this task? Here, we show that in the absence of physical heuristics, the inference of optimal quantum models cannot be computed efficiently (unless P = NP). This result illuminates rigorous limits to the extent to which computers can be used to further our understanding of nature. DOI: 10.1103/PhysRevA.97.020103 A characterization of a physical experiment is always at To describe the experiment quantum mechanically we need least twofold. On the one hand, we have a description to translate the descriptions Sx and Oyz into quantum states ρ and measurement operators E . This corresponds to the S = (description of the state) x yz task of modeling. The assignment of matrices to Sx and Oyz of the state of the physical system. For instance, S can contain a must be such that the quantum mechanical predictions are few paragraphs of text with detailed instructions for preparing compatible with the previously measured data fxyz. By Born’s that state experimentally in the laboratory, or for finding it in rule, tr(ρx Eyz) is the probability for measuring outcome z nature. if we measure state Sx with the measurement My . Hence, The second part of the characterization of an experiment is achieving compatibility between the theoretical model ρx ,Eyz the description of the measurement that is performed. As for on the one hand and the experimental description Sx ,Oyz on the state, the measurement may be described in terms of a short the other hand requires searching for states and measurements text, satisfying tr(ρx Eyz) ≈ fxyz for all (x,y,z) ∈ . Here, ⊆ × × M = [X] [Y ] [Z] marks the particular combinations (x,y,z) (description of the measurement). that we have measured experimentally. Combinations in the M may be a complete manual for constructing the measure- complement (x,y,z) ∈ c are unknown. A common pitfall to ment device we use. avoid is overfitting, that is, finding an excessively complicated Both M and S can specify temporal and spatial informa- model that perfectly fits the data but has no predictive power tion, e.g., the desired state is the state resulting from a particular over future observations. To avoid overfitting we need to search initial state after letting it evolve for 1 μs. Every experimental for the lowest-dimensional model satisfying tr(ρx Eyz) ≈ fxyz. paper must provide M and S. In fact, if we placed no restriction on the dimension, then Performing the measurement M results in a measurement we could fit every data set exactly with a finite-dimensional outcome. We denote by Z the number of different measurement quantum model that does not allow for the prediction of future outcomes. Each of the outcomes may again be characterized measurement outcomes. For instance, we could fit the mea- in terms of a few paragraphs of text, sured data with an X-dimensional model where ρx =|xx| and E = X f |xx|. Indeed, tr(ρ E ) = f .Onthe O = yz x=1 xyz x yz xyz z (description of zth measurement outcome), other hand, if a subsystem structure (e.g., two independent for all z ∈ [Z]:={1,...,Z}. Here, we assume without loss of parties Alice and Bob) is imposed, then there are circumstances generality that the description Oz also specifies M, i.e., it both where data sets cannot be modeled by finite-dimensional fully specifies the measurement device and the way it signals quantum models [1,2]. outcome z has been measured to the observer. In the remainder we are going to assume that the em- Oftentimes we do not only consider a single state S and pirical frequencies fxyz are equal to the probabilities pxyz O S a single measurement (O ) ∈ but X states (S ) ∈ and Y for measuring outcome yz given that we prepared x and z z [Z] x x [X] M measurements (O ) (y ∈ [Y ]). For instance, we could be measured y . This condition is met if we can measure states yz z∈[Z] S M interested in measuring the spin of an electron in different x with measurements y an unbounded number of times →∞ directions and at different times. Repeatedly measuring the (Nxy ). We will see that inference is NP-hard even in this noiseless setting where we want to solve state Sx with the measurement My we are able to collect Z empirical frequency distributions (fxyz)z=1 for that particular sequence of measurements, that is, fxyz = {z|xy}/Nxy, where minimize d N denotes the number of times we measure S with M and xy x y ∃ d where {z|xy} denotes the number of times we see outcome such that -dimensional states and measurements Oyz during these runs of the experiment. satisfying pxyz = tr(ρx Eyz) ∀(x,y,z) ∈ . (1) 2469-9926/2018/97(2)/020103(4) 020103-1 ©2018 American Physical Society CYRIL J. STARK PHYSICAL REVIEW A 97, 020103(R) (2018) We call problem (1) MinDim; it describes the task of learning Communication, we prove the latter by reduction from 3col. effective quantum models from experimental data. Our result Thus, we are arriving at our main result, Theorem 1. that MinDim is NP-hard implies that computers are not capable Theorem 1. MinDim is NP-hard. of computing optimal quantum models describing general Every experiment can be described in terms of (Sx )x and experimental observations (unless P = NP). (O)yz. Therefore, problem (1) does not make any assump- NP-hardness is a term from computational complexity tions about the underlying quantum model. Often, however, theory which aims at classifying problems according to their we accept some side information about the physical sys- complexity. The relevant complexity measure depends on the tem we wish to analyze. A common postulate is that we particular application. Here, we focus on time complexity measure a global state with local measurements [4–7]. In which measures the time it takes to solve a problem on this setting we want to solve the following modification of a computer (deterministic Turing machine). A particularly MinDim, important family of problems are decision problems. These are problems whose solution is either yes or no. The 3-coloring minimize d of graphs is a famous example. In 3-coloring (3col) we are such that ∃ a d2-dimensional state ρ and d-dimensional given a graph with vertices specified by a vertex set V and with edges specified by an edge set E. Our task is to decide measurements (Eyz)z and (Fyz)z satisfying whether or not it is possible to assign colors red, green,orblue pyzy z = tr(ρEyz ⊗ Fy z ) ∀(yzy z ) ∈ (2) to vertices v ∈ V in such a way that vertices v,v are colored differently whenever the edge (v,v ) with endpoints v,v is an (for some ⊆ [Y ] × [Z] × [Y ] × [Z ]). We are referring to element of E. In this example, the specification of V and E problem (2) in terms of MinDim(AB); the label (AB) references forms the problem instance and the criterion for the solution yes two parties, usually called Alice and Bob. Here, we prove NP- (i.e., yes, this graph is 3-colorable) is the so-called acceptance hardness of MinDim(AB) by showing that the natural decision condition. A decision problem is specified by an acceptance problem Dim-3(AB) (see the Supplemental Material [8]) of condition and by a set of problem instances. MinDim(AB) is NP-hard. The complexity classes P and NP have been introduced to Theorem 2. MinDim(AB) is NP-hard. classify problems according to their complexity. The complex- Theorems 1 and 2 assume that the measurement probabil- ity class P is the set of all decision problems whose complexity ities pxyz and pyzy z are known exactly. Hence, Theorems 1 is a polynomial in the size of the problem instances (e.g., and 2 do not allow one to draw rigorous conclusions about the number of vertices in case of 3col). The class NP is situations where pxyz are only known approximately. When the set of problems with the following property. Every yes does a physical theory qualify to be a good physical theory? instance admits a proof that can be checked in polynomial Answers provided are sometimes vague. However, there is time. For example, in the case of 3col, we can prove that a a consensus that predictive power is a necessary criterion a graph is 3-colorable by providing an explicit 3-coloring of good physical theory needs to satisfy. This criterion is satisfied that graph; the correctness of that coloring can be verified by if models drawn from that theory (e.g., quantum theory) checking that for all (v,v ) ∈ E, the vertices v and v are colored allow for the prediction of future measurement outcomes, differently.