Critical Points in Quantum Generative Models

1 INTRODUCTION . 1.1 MOTIVATION . One of the great successes of neural networks is the efficiency at which they are trained via gradientbased methods . Though training algorithms often involve the optimization of complicated , nonconvex , high-dimensional functions , training via gradient descent in many contexts manages to converge to local minima that are good approximations of the global minimum in loss function value . This phenomenon has begun to be understood in the context of random matrix theory , particularly when applied to dense classifiers ( Choromanska et al. , 2015 ; Chaudhari & Soatto , 2017 ) . Quantum generative models hold great promise in their ability to sample from probability distributions out of the reach of classical models ( Arute et al. , 2019 ; Gao et al. , 2021 ) . Though deep quantum generative models are believed to be difficult to train due to vanishing gradients ( McClean et al. , 2018 ; Cerezo et al. , 2021 ; Marrero et al. , 2020 ) , the situation for shallow models is murkier . Some shallow quantum models have empirically shown great promise in being trainable ( Wiersema et al. , 2020 ; Kim et al. , 2020 ; Kim & Oz , 2021 ) , while others have empirically been shown to suffer from poor distributions of local minima ( Kiani et al. , 2020 ; Campos et al. , 2021 ) . Numerically , all of these models have been seen to experience a phase transition in trainability : below some critical depth , local minima are poor approximators of the global minimum . Above this critical depth , they are good approximators . This transition has been poorly understood analytically , as typically the distribution of local minima monotonically improves as the size of the model increases ( Choromanska et al. , 2015 ; Chaudhari & Soatto , 2017 ) . 1.2 OUR CONTRIBUTIONS . In this work , we are the first to analytically show the presence of a computational phase transition in the training of a certain class of quantum generative models . To achieve this , we first show that this class of randomized quantum models is approximated in distribution by a Wishart random field on the hypertorus . We are then able to use techniques from Morse theory to exactly calculate the distribution of local minima ( and general critical points ) of this random field . Finally , we analyze this distribution in the limit of large model size , and analytically show the presence of this trainability phase transition . Roughly , we show that in this limit the expected density of local minima for a model with p parameters and Hilbert space dimension ∼ m ( exponential in the problem size ) at loss value 0 ≤ E ≤ 12 follows a generalization of the beta distribution ( Gordy , 1998 ) : E [ Crt0 ( E ) ] ∼ e−mEEm− p 2 ( 1− 2E ) p . ( 1 ) This distribution experiences a transition in behavior at p = 2m : when p < 2m , local minima are exponentially concentrated ( i.e . with width m−1 ) far away from the global minimum E = 0 , implying poor optimization performance in this regime . When p ≥ 2m , this distribution is exponentially concentrated at E = 0 , implying good optimization performance . We also verify our results numerically , demonstrating this concentration of minima even at small problem sizes . For the class of quantum generative models we consider , our results mirror the empirical results of Kiani et al . ( 2020 ) ; Campos et al . ( 2021 ) in that only unreasonably overparameterized quantum models have good local minima . Though these results are pessimistic , we emphasize here that our results only apply to a certain class of quantum generative models . We are also able to give a heuristic explanation based on our proof techniques as to how one may be able to construct models of a reasonable size that are still trainable at the expense of computational overhead in implementing the model , as seen empirically in Wiersema et al . ( 2020 ) ; Kim et al . ( 2020 ) ; Kim & Oz ( 2021 ) . 2 PRELIMINARIES . 2.1 QUANTUM GENERATIVE MODELS . A quantum system of size n is naturally represented by a quantum state , which is a normalized vector |ψ〉 ∈ C2n . Here , we use the typical physics notation |·〉 to denote a vector , instead of ( say ) ψ , when we are describing a quantum state . A quantum state in C2n can be considered a generalization of probability distributions over 2n states ( i.e . over states described by n bits ) , where the norm squared of entries of |ψ〉 give the distribution . The task in quantum generative modeling is to prepare a state |ψ〉 that is close under some metric to a given target state |ψtarget〉 . As operations that map probability distributions to probability distributions are naturally described by stochastic matrices , operations that map quantum states to quantum states are naturally described by unitary matrices ; equivalently , they are described by the matrix exponentials of 2n × 2n skewHermitian matrices . Multiple quantum operations can then be described by the sequential matrix multiplication of various matrix exponentials . It is then apparent that one can construct a quantum generative model by parameterizing these matrix exponentials , giving rise to the layered structure : |θ〉 ≡ q∏ i=1 Ui ( θ ) |ψ0〉 ≡ q∏ i=1 e−iθiQi |ψ0〉 , ( 2 ) where here q is the depth of the model and Qi are fixed Hermitian matrices . Typically , q is polynomial in n , i.e . logarithmic in the dimension of the initial state vector |ψ0〉 . In most quantum generative models , various θi are completely dependent , e.g . θi = θi+5 for all i ( Romero et al. , 2018 ) . For simplicity , we assume throughout this work that each independent parameter appears a constant number r times in the model , and the total number of independent parameters is given by p = q/r . Though the linear nature of the model may be surprising , this model structure ( for large enough depth q ) is known to be a universal approximator of all ( pure ) quantum states , even for a fixed number of allowed Qi ( Solovay , 1995 ; Kitaev , 1997 ) . For completely general ( i.e . dense ) Qi in equation 2 , the model is not efficient to implement on a quantum computer . Thus , due to their efficiency in physical implementation , these Qi are typically taken to be members of the n-qubit Pauli group Pn , which is a generating subset of the additive group of all n × n Hermitian matrices . The Pauli group is also convenient to study analytically , as is the normalizer of the group ( called the Clifford group ) . The assumption that each Qi is a Pauli operator also allows us to use a single parameter θi for each layer of the model without loss of generality ; in principle , more parameters can describe each layer by parameterizing sparse Qi . However , as the Pauli group generates Hermitian matrices , this is a special case of the class of models we consider here ( at the expense of larger q and new dependencies among the θi ) . In quantum generative modeling , we are often not given the target state |ψtarget〉 directly . Instead , we are in many cases given a 2n × 2n Hermitian matrix H ( called the problem Hamiltonian ) where |ψtarget〉 is the eigenvector associated with the smallest eigenvalue of H ( called the ground state ) . In this formulation , optimization proceeds via the minimization of : FVQA ( θ ) = 〈θ|H |θ〉 . ( 3 ) In physics notation , 〈θ| is the conjugate transpose of the complex vector |θ〉 . Assuming no degeneracies in the eigenspectrum ofH and a sufficiently expressive model , the minimizer |θ∗〉 of equation 3 is the ground state of H , up to an overall phase due to the quadratic nature of equation 3 . Assuming H is efficiently expressible as the weighted sum of O ( poly ( n ) ) Pauli matrices , equation 3 and its gradients can be efficiently measured on a quantum computer ( Peruzzo et al. , 2014 ; Romero et al. , 2018 ) . This and similar formulations of quantum generative modeling with equation 3 as the loss function are called variational quantum algorithms ( VQAs ) ( Peruzzo et al. , 2014 ) . Generally , the goal of these algorithms is to find the state |θ〉 that optimizes equation 3 , up to potentially some constant additive error in loss . Though there are other formulations of quantum generative modeling , we here focus on VQAs as they do not require coherent access to data |ψtarget〉 , which is generally believed to be difficult ( Aaronson , 2015 ) . Typically , models in VQAs come in one of two flavors : Hamiltonian agnostic models , and Hamiltonian informed models . Hamiltonian agnostic models are constructed such that the Qi present in the model definition are independent of H , and are generally more efficient to implement . This is most analogous to the case in classical generative modeling , where the model structure is usually independent from the specific choice of data distribution . We prove the existence of a computational phase transition for a class of Hamiltonian agnostic models in this work . We also give some heuristic and numerical evidence that a similar transition exists for Hamiltonian informed ansatzes , but that the trainable phase is more easily reached . 2.2 RANDOM FIELDS ON MANIFOLDS . As in previous results on the loss landscapes of machine learning models ( Choromanska et al. , 2015 ; Chaudhari & Soatto , 2017 ) , we will map the distribution of a randomized class of quantum generative models to a random field on a manifold . This will then enable us to use standard mathematical techniques to study the distribution of critical points of the model . Though they can be expressed in many ways , here we will be interested in random fields of the form : FRF ( σ ) ∝ Λ∑ i1 , ... , ir , i′1 , ... , i ′ r=1 σi1 . . . σirJi1 , ... , ir , i′1 , ... , i′rσi′1 . . . σi′r . ( 4 ) Here , σ ∈M is some point on a manifold , and J is a random variable . In the context of most studies of machine learning loss landscapes , M is typically the hypersphere and J a symmetric matrix of i.i.d . Gaussian random variables , i.e . a Gaussian orthogonal ensemble ( GOE ) matrix . We will instead find that VQAs are naturally described as Wishart hypertoroidal random fields ( WHRFs ) . For these models , the manifold M is a tensor product embedding of the hypertorus into exponentially large Euclidean space ; that is , points on this embedding are described by the Kronecker product : σ = ⊗ i ( cos ( θi ) sin ( θi ) ) ( 5 ) for angles −π ≤ θi ≤ π . Furthermore , in these models , J is drawn from a normalized complex Wishart distribution . The complex Wishart distribution is a natural multivariate generalization of the gamma distribution , and is given by the distribution of the square of a complex Gaussian random matrix . Specifically , forX ∈ Cn×m a matrix with i.i.d . complex Gaussian columns with covariance matrix Σ , the matrix W = 1 m X ·X† ( 6 ) is normalized complex Wishart distributed with scale matrix Σ and m degrees of freedom . We will find that the degrees of freedom m will greatly affect the distribution of local minima of the WHRF , and thus also of the class of quantum generative models that we consider .

Summary: The work attempts to connect the optimization of quantum generative models with random fields on manifolds. Some theoretical and numerical analyses are provided.   The major idea is to apply random matrix theory for variational quantum circuits. When the authors provide some numerical experiments, overall the validation is not very strong and could be improved.   The connection between applying random fields and quantum circuit training is not very direct based on the current writing. Some related references in the field are missing. (e.g., [1,2]).  ***  **Reference**  1. Random matrix spectral form factor of dual-unitary quantum circuits." Communications in Mathematical Physics 387.1 (2021): 597-620.  2. Abrupt transitions in variational quantum circuit training." Physical Review A 103.3 (2021): 032607.