Impact of Barren Plateaus on the Hessian and Higher Order Derivatives
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Impact of Barren Plateaus on the Hessian and Higher Order Derivatives
M. Cerezo1, 2 and Patrick J. Coles1
1
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
2
Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, USA
The Barren Plateau (BP) phenomenon is an issue for certain quantum neural networks and
variational quantum algorithms, whereby the gradient vanishes exponentially in the system size n.
The question of whether high-order derivative information such as the Hessian could help escape a BP
was recently posed in the literature. Here we show that the elements of the Hessian are exponentially
suppressed in a BP, so estimating the Hessian in this situation would require a precision that scales
exponentially with n. Hence, Hessian-based approaches do not circumvent the exponential scaling
associated with BPs. We also show the exponential suppression of higher order derivatives. Hence,
BPs will impact optimization strategies that go beyond (first-order) gradient descent.
arXiv:2008.07454v1 [quant-ph] 17 Aug 2020
I. Introduction was to compute the Hessian H of the cost function, and
the claim was that taking a learning rate proportional to
the inverse of the largest eigenvalue of the Hessian leads
Training parameterized quantum circuits is a promis-
to an optimization method that could escape the barren
ing approach for quantum computing in the Noisy In-
plateau.
termediate Scale Quantum (NISQ) era. This strategy is
employed in two closely related paradigms: Variational The question of whether higher-order derivative infor-
Quantum Algorithms (VQAs) for chemistry, optimiza- mation (beyond the first-order gradient) is useful for es-
tion, and other applications [1–12], and Quantum Neural caping a barren plateau is interesting and is the subject of
Networks (QNNs) for classification applications [13–16]. our work here. Our main results are presented here in the
In both cases, one utilizes a quantum computer to ef- form of two propositions and corollaries. First, we show
ficiently evaluate a cost (or loss) function C(θ) or its that the matrix elements Hij of the Hessian are exponen-
gradient ∇C(θ), while employing a classical optimizer tially vanishing when the cost exhibits a barren plateau.
to train the parameters θ of a parameterized quantum This implies that the calculation of Hij requires expo-
circuit V (θ). While many novel VQAs and QNNs have nential precision. In our second result we show that the
been developed, more rigorous scaling analysis is needed magnitude of any higher-order partial derivative of the
for these architectures. cost will also be exponentially small in a barren plateau.
Our results suggest that optimization methods that use
One of the few known results is the so-called bar- higher-order derivative information, such as the Hessian,
ren plateau phenomenon [17–20], where the cost func- will also face exponential scaling, and hence do not cir-
tion gradient vanishes exponentially with the system size. cumvent the scaling issues arising from barren plateaus.
This can arise due to deep unstructured ansatzes [17, 19],
global cost functions [18, 19], or noise [20]. Regardless
of the origin, when a cost landscape exhibits a barren II. Preliminaries
plateau, one requires an exponential precision to deter-
mine a minimizing direction in order to navigate the land-
scape. Since the standard goal of quantum algorithms is In what follows, we consider the case when the cost
polynomial scaling with the system size (in contrast to can be expressed as a sum of expectation values:
the exponential scaling of classical algorithms), the expo- N
nential scaling due to barren plateaus can destroy quan- C(θ) =
X
Cx , with Cx = Tr[Ox V (θ)ρx V † (θ)] , (1)
tum speedup. Hence, the study and analysis of barren
x=1
plateaus should be viewed as a fundamental step in the
development of VQAs and QNNs to guarantee that they where {ρx } is a set (of size N ) of input states to the
can, in fact, provide a speedup over classical algorithms. parameterized circuit V (θ). In order for this cost to be
Recently there have been multiple strategies proposed efficiently computable, the number of states in the input
for avoiding barren plateaus such as employing local set should grow at most polynomially with the number
cost functions [18], pre-training [21], parameter corre- of qubits n, that is, N ∈ O(poly(n)). In the context
lation [22], layer-by-layer training [23], and initializing of QNNs, the states {ρx } can be viewed as training data
layers to the identity [24]. These strategies are aimed at points, and hence (1) is a natural cost function for QNNs.
either avoiding or preventing the existence of a barren In the context of VQAs, one typically chooses N = 1,
plateau, and they appear to be promising, with more re- corresponding to a single input state. In this sense, the
search needed on their efficacy on general classes of prob- cost function in (1) is general enough to be relevant to
lems. In a recent manuscript [25], an alternative idea was both QNNs and VQAs.
proposed involving a method for actually training inside Let θi be an angle which parameterizes a unitary in
and escaping a barren plateau. Specifically, the proposal V (θ) as e−iθi σi /2 , with σi a Hermitian operator with2
eigenvalues ±1. Then, the partial derivative ∂C(θ) ∂θi = From ChebyshevâĂŹs inequality we can bound the prob-
∂i C(θ) can be computed via the parameter shift rule [26, ability that the cost derivative deviates from its mean
27] as value (of zero) as
1 (1)
(− 1 )
Varθ [∂i C]
∂i C(θ) = C θ i , θi 2 − C θ i , θi 2 , (2) Pr (|∂i C| > c) 6 , (9)
2 c2
where we define for all c > 0, and for all i. Then, let E±be defined
as the
(δ) (± 12 )
θi = θi + δπ. (3) event that |∂i C(θ ± )| > c, where θ ± = θ j , θj . Note
Note that the parameter shift rule in (2) allows one to that the set of events where |Hij | > c is a subset of the
exactly write the first-order partial derivative as a dif- set E+ ∪ E− . Then, from the union bound and Eq. (9) we
ference of cost function values evaluated at two different can recover (7) as follows:
points.
Pr(|Hij | > c) 6 Pr(E+ ∪ E− ) (10)
6 Pr(E+ ) + Pr(E− ) (11)
III. Hessian matrix elements Varθ+ [∂i C] Varθ− [∂i C]
6 + (12)
c2 c2
Let us now state our results for the Hessian. The Hes- 2Varθ [∂i C]
sian H of the cost function is a square matrix whose = , (13)
c2
matrix elements are the second derivatives of C(θ), i.e.,
where we used the fact that h·iθ = h·iθ± .
∂ 2 C(θ)
Hij = = ∂i ∂j C(θ) . (4)
∂θi ∂θj When the cost exhibits a barren plateau as in [17–
19], the variance of the cost function partial derivative
Reference [25] noted that the matrix elements of the Hes- Varθ [∂i C] is exponentially vanishing as
sian can be written according to the parameter shift rule.
Namely, one can first write Varθ [∂i C] 6 F (n) , with F (n) ∈ O(1/bn ). (14)
1h
(1)
(− 1 )
i
Hij = ∂i C θ j , θj 2 − ∂i C θ j , θj 2 (5) for some b > 1. Then, the following corollary holds.
2
and then apply the parameter shift rule a second time: Corollary 1. Consider the bound in Eq. (10) of Proposi-
tion 1. If the cost exhibits a barren plateau, such that (14)
1h (1) (1) (− 1 ) (− 1 ) holds, then the matrix elements of the Hessian are expo-
Hij = C θ ij , θi 2 , θj 2 + C θ ij , θi 2 , θj 2 (6)
4 nentially vanishing since
i
(1) (− 1 ) (− 1 ) (1)
−C θ ij , θi 2 , θj 2 − C θ ij , θi 2 , θj 2 . 2F (n)
Pr(|Hij | > c) 6 , (15)
c2
Now, the second derivatives of the cost can be expressed
as a sum of cost functions being evaluated at (up to) four where F (n) ∈ O(1/bn ) for some b > 1.
points.
From the parameter shift rule we can then derive the The proof follows by combining (7) and (14). Corol-
following bound on the probability that the magnitude of lary 1 shows that when the cost landscape exhibits a
the matrix elements |Hij | are larger than a given c > 0. barren plateau, the matrix elements of the Hessian are
exponentially vanishing with high probability. This im-
Proposition 1. Consider a cost function of the plies that any algorithm that requires the estimation of
form (1), for which the parameter shift rule of (2) holds. the Hessian will requires a precision that grows exponen-
Let Hij be the matrix elements of the Hessian as defined tially with the system size.
in (5). Then, assuming that h∂i Ciθ = 0, the following
inequality holds for any c > 0,
2Varθ [∂i C] IV. Higher order partial derivatives
Pr(|Hij | > c) 6 . (7)
c2
Let us now analyze the magnitude of higher order par-
Here Varθ [∂i C] = h(∂i C)2 iθ − h∂i Ci2θ , where the expec- tial derivatives in a barren plateau. We use the following
tation values are taken over θ. notation for the |α|th-order derivative
Proof. Equation (5) implies that the magnitudes of the
Dα C(θ) = ∂α1 ∂α2 · · · ∂α|α| C(θ) , (16)
Hessian matrix elements are bounded as
1
(1)
(− 1 )
where α is an |α|-tuple. Since one can take the deriva-
|Hij | 6 ∂i C θ j , θj 2 + ∂i C θ j , θj 2 . (8)
2 tive with respect to the same angle multiple times, we3
M
Y X
Wω = d(ωl ,Nl ) such that |Wω | = 2|α| . (19)
l=1 ω
Also, ω = (ω1 , . . . , ωD ), where ωl ∈ {0, ±1} if Nl is even,
and ωl ∈ {± 12 , ± 32 } if Nl is odd. Additionally, the coeffi-
cients dωl ,Nk can be obtained from the Pascal tree which
we introduce in Fig. 1. In the Appendix we provide ad-
ditional details regarding the coefficients d(ωl ,Nl ) and the
Pascal tree.
From (18) we obtain that the (|α| + 1)th-order deriva-
tive, which we denote as ∂i Dα C(θ) = Di,α C(θ), is ob-
tained as the sum of (up to) 2|α| partial derivatives:
1 X
FIG. 1. The Pascal tree. a) The Pascal tree can be ob- Di,α C(θ) = |α| Wω ∂i C(Θ, Θ(ω) ) . (20)
tained by modifying how a Pascal triangle is constructed. In 2 ω
a Pascal triangle each entry of a row is obtained by adding to-
gether the numbers directly above to the left and above to the Since one has to individually evaluate each term in (20)
right, with blank entries considered to be equal to zero. The and since there are up to 2|α| terms, we henceforth as-
entries of a Pascal tree are obtained following the aforemen- sume that |α| ∈ O(log(n)). This guarantees that the
tioned rule, with the additional constraint that the width of computation of Di,α C(θ) leads to an overhead which is
the triangle is restricted to always being smaller than a given (at most) O(poly(n)).
even number. Moreover, once an entry in a row is outside The following proposition, which generalizes Proposi-
the maximum width, its value is added to the central entry tion 1, allows us to bound the probability that the mag-
in that row (see arrows). Here the maximum width is four. nitude of Di,α C(θ) is larger than a given c > 0.
b) The coefficients d(ωl ,Nl ) in (19) can be obtained from the
Pascal tree of (a) by adding signs to the entries of the tree. Proposition 2. Consider a cost function of the
As schematically depicted, all entries in a diagonal going from form (1), for which the parameter shift rule of (2) holds.
top left to bottom right have the same sign, with the first en- Let Di,α C(θ) be a higher order partial derivative of the
try in the first row having a positive sign. Here, each row cost as defined in (16). Then, assuming that h∂i Ciθ = 0,
corresponds to a given Nl , while entries in a row correspond the following inequality holds for any c > 0,
to different values of ωl , with ωl ∈ {0, ±1} if Nl is even, and
ωl ∈ {± 12 , ± 23 } if Nl is odd. For instance, d(− 1 ,5) = −12. 2|α| Varθ [∂i C]
2
Pr( Di,α C(θ) > c) 6 . (21)
c2
Proof. From Eq. (20) we can obtain the following bound
define the set Θ (of size M = |Θ|) as the set of dis-
tinct angles with respect to which we take the partial
derivative. Similarly, let Θ be the compliment of Θ, so 1 X
Di,α C(θ) 6 |Wω | ∂i C(Θ, Θ(ω) ) . (22)
that Θ ∪ Θ = θ. Then, for any Θk ∈ Θ we define Nk 2|α| ω
PM
as the multiplicity of Θk in α such that k=1 Nk = |α|.
Since the cost function and any of its higher order partial Let us define Eω as the event that ∂i C(Θ, Θ(ω) ) > c.
derivatives are continuous function of the parameters (as Since (14) holds, then the following chain of inequalities
can be seen below via multiple applications of the param- holds
eter shift rule), one can extend Clairaut’s Theorem [28] !
to rewrite i,α
[
Pr( D C(θ) > c) 6 Pr Eω (23)
N1 NM ω
Dα C(θ) = ∂Θ1
· · · ∂ΘM
C(θ) . (17) X
6 Pr (Eω ) (24)
Then, applying the parameter shift rule |α| times we ω
find that the |α|-order partial derivative can be expressed X Var(Θ,Θ(ω) ) [∂i C]
as a summation of cost functions evaluated at (up to) 2|α| 6 (25)
ω
c2
points as
|α|
2 Varθ [∂i C]
1 X 6 , (26)
Dα C(θ) = Wω C(Θ, Θ (ω)
). (18) c2
2|α| ω where we invoked the union bound, and where we recall
that h·iθ = h·i(Θ,Θ(ω) ) , ∀ω. In addition, for (26) we used
(ω ) (ω )
Here we defined Θ(ω) = (Θ1 1 , . . . , ΘM M ), with the fact that the summation in (25) has at most 2|α|
(ω )
Θk k = Θk + ωk π defined analogously to (3), and where terms.4
Then, if the cost function exhibits a barren plateau, V. Discussion
the following corollary follows.
In this work, we investigated the impact of barren
Corollary 2. Consider the bound in Eq. (21) of Proposi- plateaus on higher order derivatives. This issue was im-
tion 2. If the cost exhibits a barren plateau, such that (14) portant in light of a recent proposal to use to higher
holds, then higher order partial derivatives of the cost order derivative information to escape a barren plateau.
function are exponentially vanishing since We considered a cost function C that is relevant to both
VQAs and QNNs, as barren plateaus are relevant to both
G(n) of these applications.
Pr( Di,α C(θ) > c) 6 , (27)
c2 Our main result was that, when a barren plateau ex-
ists, the Hessian and other high order partial derivatives
where G(n) ∈ O(1/q n ) for some q > 1. of C are exponentially vanishing in n with high probabil-
ity. Our proof relied on the parameter shift rule, which
Proof. Combining (14) and (21) leads to we showed can be applied iteratively to relate higher or-
der partial derivatives to the first order partial derivative
2|α| F (n) (analogous to what Ref. [25] did for the Hessian). Hence,
Pr( Di,α C(θ) > c) 6 . (28) the parameter shift rule allowed us to state the vanishing
c2
of higher order derivatives as essentially a corollary of the
vanishing of the first order derivative. We remark that
Then, let us define G(n) = 2|α| F (n). Since |α| ∈
iterative applications of the parameter shift rule led us
O(log(n)), and F (n) ∈ O(1/bn ), then we know that there
to a mathematically interesting construct that we called
exists κ, κ0 , and n0 such that ∀n > n0 we respectively
0 the Pascal tree, depicted in Fig. 1.
have 2|α| 6 nκ and F (n) 6 bκn . Combining these two Our results imply that estimating higher order par-
results we find tial derivatives in a barren plateau is exponentially hard.
Hence, any optimization strategy that requires informa-
κ0 nκ κ0
G(n) 6 n
= L(n) , ∀n > n0 , (29) tion about partial derivatives that go beyond first-order
b b (such as the Hessian) will require a precision that grows
exponentially with n. We therefore surmise that, by
where L(n) = (n − κ logb (n)). Equation (29) shows that
themselves, optimizers that go beyond first order gra-
G(n) ∈ O(1/bL(n) ). Then, since
dient descent do not appear to be a feasible solution to
the barren plateau problem. More generally, our results
L(n) suggest that it is better to develop strategies that avoid
lim = 1, (30)
n→∞ n the appearance of the barren plateau altogether, rather
than to try to escape an existing barren plateau.
we have L(n) ∈ Ω(n), meaning that there exists a κ b>0 Acknowledgements.—We thank Kunal Sharma for
and nb0 such that ∀n > n b0 , we have L(n) > κ bn. The helpful discussions. Research presented in this article
0
latter implies G(n) 6 bκκb n for all n > max{n0 , n
b0 }, which was supported by the Laboratory Directed Research and
n κ
means that G(n) ∈ O(1/q ) where q = b . Also, q > 1
e Development program of Los Alamos National Labora-
follows from b > 1 and κ e > 0. tory under project number 20180628ECR. MC was also
supported by the Center for Nonlinear Studies at LANL.
PJC also acknowledges support from the LANL ASC Be-
Corollary 2 shows that, in a barren plateau, the mag- yond Moore’s Law project. This work was also supported
nitude of any efficiently computable higher order par- by the U.S. Department of Energy (DOE), Office of Sci-
tial derivative (i.e., any partial derivative where |α| ∈ ence, Office of Advanced Scientific Computing Research,
O(log(n))) is exponentially vanishing in n with high under the Accelerated Research in Quantum Computing
probability. (ARQC) program.
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