Abstract
Many experimental proposals for noisy intermediate scale quantum devices involve training a parameterized quantum circuit with a classical optimization loop. Such hybrid quantumclassical algorithms are popular for applications in quantum simulation, optimization, and machine learning. Due to its simplicity and hardware efficiency, random circuits are often proposed as initial guesses for exploring the space of quantum states. We show that the exponential dimension of Hilbert space and the gradient estimation complexity make this choice unsuitable for hybrid quantumclassical algorithms run on more than a few qubits. Specifically, we show that for a wide class of reasonable parameterized quantum circuits, the probability that the gradient along any reasonable direction is nonzero to some fixed precision is exponentially small as a function of the number of qubits. We argue that this is related to the 2design characteristic of random circuits, and that solutions to this problem must be studied.
Introduction
Rapid developments in quantum hardware have motivated advances in algorithms to run in the socalled noisy intermediate scale quantum (NISQ) regime^{1}. Many of the most promising applicationoriented approaches are hybrid quantum–classical algorithms that rely on optimization of a parameterized quantum circuit^{2,3,4,5,6,7,8}. The resilience of these approaches to certain types of errors and high flexibility with respect to coherence time and gate requirements make them especially attractive for NISQ implementations^{3,9,10,11}.
The first implementation of such algorithms was developed in the context of quantum simulation with the variational quantum eigensolver^{2,3}. This algorithm has been successfully demonstrated on a number of experimental setups with extensions to excited states and other forms of incoherent error mitigation^{2,9,12,13,14,15,16}. Since then, the quantum approximate optimization algorithm was developed in a similar context to address hard optimization problems^{5,17,18,19}. This algorithm has also been demonstrated on quantum devices^{20}. These approaches have even been extended to both quantum machine learning and error correction^{6,7,20,21,22,23}.
While the precise formulation of these methods and their domains of applicability differ considerably, they typically tend to rely upon the optimization of some parameterized unitary circuit with respect to an objective function that is typically a simple sum of Pauli operators or fidelity with respect to some state. This framework is reminiscent of the methodology of classical neural networks^{23,24}. As with any nonlinear optimization, the choice of both the parameterization and the initial state is important. In quantum simulation, there is often a choice inspired by physical domain knowledge^{3,17,25,26,27,28,29}. However, in all domains of applicability, there have been implementations that utilize parametrized random circuits of varying depth^{7,13,21,23,30}. Within quantum simulation that approach has been referred to as a “hardware efficient ansatz”^{13}. This is in contrast to the previous proposals, such as the variational quantum eigensolver^{2,3,9}, which used parametrized structured circuits inspired by the problem at hand, such as unitary coupled cluster.
When little structure is known about the problem, or constraints of the existing quantum hardware may prevent utilizing that structure, choosing a random implementable circuit seems to provide an unbiased choice. One might also expect, based on recent experimental designs for “quantum supremacy”, that random quantum circuits are a powerful tool for such a task^{31}. Also, despite concerns about gradientbased methods in classical deep neural networks^{32,33,34}, they are successful^{24}, even if using random initialization^{33,35}. However, in the quantum case one must remember that the estimation of even a single gradient component will scale as O(1/ε^{α}) for some small power α^{36} as opposed to classical implementations where the same is achieved in O(log(1/ε)) time, where ε is the desired accuracy in the gradient that is inevitably tied to its magnitude.
We will present results related to random quantum circuits in the context of the exponential dimension of Hilbert space and gradientbased hybrid quantum–classical algorithms. A cartoon depiction of this is given in Fig. 1. We show that for a large class of random circuits, the average value of the gradient of the objective function is zero, and the probability that any given instance of such a random circuit deviates from this average value by a small constant ε is exponentially small in the number of qubits. This can be understood in the geometric context of concentration of measure^{37,38,39} for highdimensional spaces. When the measure of the space concentrates in this way, the value of any reasonably smooth function will tend towards its average with exponential probability, a fact made formal by Levy’s lemma^{40}. In our context, this means that the gradient is zero over vast reaches of quantum space. The region where the gradient is zero does not correspond to local minima of interest, but rather an exponentially large plateau of states that have exponentially small deviations in the objective value from the average of the totally mixed state. We argue that the depth of circuits which achieve these undesirable properties are modest, requiring only O(n^{1/d}) depth circuits on a d dimensional array, and numerically evaluate the constant factors one expects to encounter for small instances of this kind. While our results highlight the importance of avoiding random initialization in parametric circuit approaches, they do not discount the value of random quantum circuits in other applications such as information security or demonstrations of quantum supremacy. We close with an outlook on how this result should shape strategies in ansatz design for scaling to larger experiments.
Results
Gradient concentration in random circuits
We will discuss random parameterized quantum circuits (RPQCs)
where U_{l}(θ_{l}) = exp(−iθ_{l}V_{l}), V_{l} is a Hermitian operator, and W_{l} is a generic unitary operator that does not depend on any angle θ_{l}. Circuits of this form are a natural choice due to a straightforward evaluation of the gradient with respect to most objective functions and have been introduced in a number of contexts already^{26,41}. Consider an objective function E(θ) expressed as the expectation value over some Hermitian operator H,
When the RPQCs are parameterized in this way, the gradient of the objective function takes a simple form:
where we introduce the notations \(U_  \equiv \mathop {\prod}\nolimits_{l = 0}^{k  1} U_l(\theta _l)W_l\), \(U_ + \equiv \mathop {\prod}\nolimits_{l = k}^L U_l(\theta _l)W_l\), and henceforth drop the subscript k from V_{k} → V for ease of exposition. Finally, we will define our RPQCs U(θ) to have the property that for any gradient direction ∂_{k}E defined above, the circuit implementing U(θ) is sufficiently random such that either U_{−}, U_{+}, or both match the Haar distribution up to the second moment, and the circuits U_{−} and U_{+} are independent.
Our results make use of properties of the Haar measure on the unitary group dμ_{Haar}(U) ≡ dμ(U), which is the unique left and rightinvariant measure such that
for any f(U) and V∈U(N), where the integration domain will be implied to be U(N) when not explicitly listed. While this property is valuable for proofs, quantum circuits that exactly achieve this invariance generically require exponential resources. This motivates the concept of unitary tdesigns^{42,43,44}, which satisfy the above properties for restricted classes of f(U), often requiring only modest polynomial resources. Suppose {p_{i}, V_{i}} is an ensemble of unitary operators, with unitary V_{i} being sampled with probability p_{i}. The ensemble {p_{i}, V_{i}} is a tdesign if
This definition is equivalent to the property that if f(U) is a polynomial of at most degree t in the matrix elements of U and at most degree t in the matrix elements of U^{*}, then averaging over the tdesign {p_{i}, V_{i}} will yield the same result as averaging over the unitary group with the respect to the Haar measure.
The average value of the gradient is a concept that requires additional specification because, for a given point, the gradient can only be defined in terms of the circuit that led to that point. We will use a practical definition that leads to the value we are interested in, namely
where p(U) is the probability distribution function of U. A review on the properties of products of independent random matrices can be found in ref.^{45}. The assumptions of independence and at least one of U_{−} or U_{+} forming a 1design in our RPQCs implies that 〈∂_{k}E〉 = 0, as shown in the Methods.
Levy’s lemma informs our intuition about the the expected variance of this quantity through simple geometric arguments. In particular, Haar random unitaries on n qubits will output states uniformly in the D = 2^{n} − 1 dimensional hypersphere. The derivative with respect to the parameters θ is Lipschitz continuous with some parameter η that depends on the operator H. Levy’s lemma then implies that the variance of measurements will decrease exponentially in the number of qubits. This intuition may be made more precise through explicit calculation of the variance, which is done in more detail in the Methods. The result to first order is
where the notation \(\langle f(u)\rangle _{U_x}\) indicates the average with u drawn from p(U_{x}), and the first case corresponds to U_{−} being a 2design and not U_{+}, the second to U_{+} being a 2design but not U_{−}, and the third to both U_{+} and U_{−} being 2designs. We emphasize the fact that this variance depends at most on polynomials of degree 2 in U and polynomials of degree 2 in U^{*}. Whereas a unitary 2design will exhibit the correct variance^{43,46}, a unitary 1design will exhibit the correct average value, but not necessarily the variance. As a result, if a circuit is of sufficient depth that for any ∂_{k}E, either U_{−} or U_{+} forms a 2design, then with high probability one will produce an ansatz state on a barren plateau of the quantum landscape, with no interesting search directions in sight.
From these results, it is clear that only either U_{+} or U_{−} needs to be sufficiently random to poison the gradient for the remainder of the circuit. For example, while it is somewhat unintuitive, even the first element of a circuit, k = 1, will have a vanishing gradient due to the circuit following it, U_{+}. Additionally, we see that there is no detailed dependence on the structure of V_{k}, other than the rate at which they help randomize the circuit, determining at what depth one expects to find an approximate 2design.
Numerical simulations
The previous section shows that for reasonable classes of RPQCs at a sufficient number of qubits and depth, one will end up on a barren plateau. Here we verify this result for even modest depth onedimensional (1D) random circuits with numerical simulations. This helps to clarify the rate of concentration for realistic circuits and shows the transition as the circuit grows in length from a single layer to a circuit demonstrating statistics analogous to a 2design.
The circuits and objective functions used in our numerical experiments begin with a layer of R_{Y}(π/4) = exp(−iπ/8 Y) gates to prevent X, Y, or Z from being an especially preferential direction with respect to gradients. Then, the circuit proceeds by a number of layers. Each layer consists of a parallel application of single qubit rotations to all qubits, given by R_{P}(θ) where P∈{X, Y, Z} is chosen with uniform probability and θ∈[0, 2π) is also chosen uniformly. This layer is followed by a layer of 1D nearest neighbor controlled phase gates, as in Fig. 2. Thus, the number of angles is the number of qubits times the number of layers.
The objective operator H is chosen to be a single Pauli ZZ operator acting on the first and second qubits, H = Z_{1}Z_{2}. The gradient is evaluated with respect to the first parameter, θ_{1,1}. This simple choice helps to extract the exponential scaling. As complex objectives can be written as sums of these operators, the results for large objectives can be inferred from these numbers. Moreover, it is clear that for any polynomial sum of these operators, the exponential decay of the signal in the gradient will not be circumvented.
From Fig. 3 we see that for a single 2local Pauli term, both the expected value of the gradient and its spread decay exponentially as a function of the number of qubits even when the number of layers is a modest linear function. Empirically for our linear connectivity, we see that value is about 10n where n is the number of qubits, following the expected scaling of O(n^{1/d}) where d is the dimension of the connectivity. For empirical reference, the expected gate depth in a chemistry ansatz such as unitary coupled cluster is at least O(n^{3}), meaning that if the initial parameters were randomized, this effect could be expected on less than 10 orbitals, a truly small problem in chemical terms. We also observe in Fig. 4 that as the number of layers increases, there is a transition to a 2design where the variance converges. This leads to a distinct plateau as the circuit length increases, where the height of the plateau is determined by the number of qubits. An additional example with an objective function defined by projection on a target state is provided as Supplementary Figures 1 and 2, showing the rapid decay of variance and similar plateaus as a function of circuit length. These results substantiate our conclusion that gradients in modestsized random circuits tend to vanish without additional mitigating steps.
Contrast with gradients in classical deep networks
Finally, we contrast our results with the vanishing (and exploding) gradient problem of classical deep neural networks^{32,33,34,47}. At least two key differences are present in the quantum case: (i) the different scaling of the vanishing gradient and (ii) the complexity of computing expected values.
The gradient in a classical deep neural network can vanish exponentially in the number of layers^{32,33}, while in a quantum circuit the gradient may vanish exponentially in the number of qubits, as shown above. In the classical case, the gradient for a weight in a neuron depends on the sum of all the paths connecting that neuron to the output, and when the weights are initialized with random values the paths have random signs which cancels the signal^{32}. The number of paths is exponential in the number of layers. In the quantum case, the number of paths is exponential in the number of gates, and also have random signs^{31}. The gradient saturates to an exponential in the number of qubits because the output state is normalized.
The estimation of the gradient for each training batch for a classical neural network is limited by machine precision and scales with O(log(1/ε)). Even if the gradient is small, as long as it is consistent enough between batches, the method may eventually succeed. On a quantum device, the cost of estimating the gradient scales as O(1/ε^{α})^{36}. For any number of measurements much lower than 1/g^{α}, where g is the norm of the gradient, a gradientbased optimization will result in a random walk. By concentration of measure, a random walk will have exponentially small probability of exiting the barren plateau. As a result, gradient descent without some additional strategy cannot circumvent this challenge on a quantum device in polynomial time.
Discussion
We have seen both analytically and numerically that for a wide class of random quantum circuits, the expected values of observables concentrate to their averages over Hilbert space and gradients concentrate to zero. This represents an interesting statement about the geometry of quantum circuits and landscapes related to hybrid quantum–classical algorithms. More practically, it means that randomly initialized circuits of sufficient depth will find relatively little utility in hybrid quantum–classical algorithms.
Historically, vanishing gradients may have played a role in the early winter of deep neural networks^{32,34,47}. However, multiple techniques have been proposed to mitigate this problem^{24,35,48,49}, and the amount of training data and computational power available has grown substantially. One approach to avoid these landscapes in the quantum setting is to use structured initial guesses, such as those adopted in quantum simulation. Another possibility is to use pretraining segment by segment, which was an early success in the classical setting^{48,50}. These or other alternatives must be studied if these ansatze are to be successful beyond a few qubits.
Methods
We explicitly show the expectation value of the gradient is 0 and that under our assumptions the variance decays exponentially in the number of qubits. By our definition of RPQCs, we have that for any specified direction ∂_{k}E, both U_{−} and U_{+} are independently distributed and either U_{−} or U_{+} match the Haar distribution up to at least the second moment (they are a 2design). The assumption of independence is equivalent to
which allows us to rewrite the expression as
We will utilize explicit integration over the unitary group with respect to the Haar measure, which up to the first moment can be expressed as^{51}
where N is the dimension of the space, typically 2^{n} for n qubits. Using this expression, one may readily verify that
which we use in the following. Now, making use of the assumption that either U_{+} or U_{−} matches the Haar measure up to the first moment (it is a 1design), we first examine the case where U_{−} is at least a 1design and find that
where we have defined \(\rho _  = U_  0\rangle \langle 0U_  ^\dagger\) and used the fact that the trace of a commutator of trace class operators is zero. In the second case, where we assume U_{+} is at least a 1design,
An advantage of the explicit polynomial formulas are that they allow an analytic calculation of the variance as well, which allows precise specification of the coefficient in Levy’s lemma. In cases where the integrals depend on up to two powers of elements of U and U^{*}, one may make use of the elementwise formula^{51}
The variance of the gradient is defined by
as we have seen above that 〈∂_{k}E〉 = 0. Through use of the above formula for integration up to the second moment of the Haar distribution, one may evaluate this expression in 3 separate cases. For simplicity and relevance, we evaluate them in the asymptotic case including only the dominant contribution as determined by the inverse dimension.
In the case where U_{−} is a 2design but not U_{+},
where \(H_u = u^\dagger Hu\) and we have defined the notation \(\langle f(u)\rangle _{U_x}\) to mean the average over u sampled from p(U_{x}). In the case where U_{+} is a 2design but not U_{−},
where \(\rho _u = u\rho u^\dagger\). Finally in the case where both U_{+} and U_{−} are 2designs
In all cases, the exponential decay of the gradient as a function of the number of qubits is evident.
Data availability
Data used to generate the above figures are available upon request from the authors.
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Acknowledgements
The authors thank Craig Gidney for helpful comments on the manuscript.
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J.R.M., S.B., V.N.S, R.B., and H.N. contributed to the formulation of ideas, calculations, and writing of the manuscript.
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McClean, J.R., Boixo, S., Smelyanskiy, V.N. et al. Barren plateaus in quantum neural network training landscapes. Nat Commun 9, 4812 (2018). https://doi.org/10.1038/s41467018070904
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