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 paper analyzes a general class of variational quantum circuits using random matrix theory, and gives a theoretical characterization of the distribution of high-quality local minima of the associated Hamiltonian loss function. A type of phase transition is predicted, where the model exhibits an abundance of high-quality local minima only at very large parameter counts, and a wide gap between global and typical local minima at more typical parameter counts. These predictions are compared with small numerical experiments, as well as previous works on this topic.