Noise-Induced Barren Plateaus in Variational Quantum Algorithms
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Noise-Induced Barren Plateaus in Variational Quantum Algorithms
Samson Wang,1, 2 Enrico Fontana,1, 3, 4 M. Cerezo,1, 5 Kunal Sharma,1, 6
Akira Sone,1, 5 Lukasz Cincio,1 and Patrick J. Coles1
1
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
2
Imperial College London, London, UK
3
University of Strathclyde, Glasgow, UK
4
National Physical Laboratory, Teddington, UK
5
Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, USA
6
Hearne Institute for Theoretical Physics and Department of Physics and Astronomy,
Louisiana State University, Baton Rouge, LA USA
Variational Quantum Algorithms (VQAs) may be a path to quantum advantage on Noisy
Intermediate-Scale Quantum (NISQ) computers. A natural question is whether noise on NISQ
devices places fundamental limitations on VQA performance. We rigorously prove a serious limita-
arXiv:2007.14384v3 [quant-ph] 8 Feb 2021
tion for noisy VQAs, in that the noise causes the training landscape to have a barren plateau (i.e.,
vanishing gradient). Specifically, for the local Pauli noise considered, we prove that the gradient
vanishes exponentially in the number of qubits n if the depth of the ansatz grows linearly with
n. These noise-induced barren plateaus (NIBPs) are conceptually different from noise-free barren
plateaus, which are linked to random parameter initialization. Our result is formulated for a generic
ansatz that includes as special cases the Quantum Alternating Operator Ansatz and the Unitary
Coupled Cluster Ansatz, among others. For the former, our numerical heuristics demonstrate the
NIBP phenomenon for a realistic hardware noise model.
I. Introduction VQAs may avoid the exponential scaling that otherwise
would result from the exponential precision requirements
One of the great unanswered technological questions of navigating through a barren plateau.
is whether Noisy Intermediate Scale Quantum (NISQ) However, these works do not consider quantum hard-
computers will yield a quantum advantage for tasks of ware noise, and very little is known about the scalability
practical interest [1]. At the heart of this discussion are of VQAs in the presence of noise. One of the main sell-
Variational Quantum Algorithms (VQAs), which are be- ing points of VQAs is noise mitigation, and indeed VQAs
lieved to be the best hope for near-term quantum ad- have shown evidence of noise resilience in the sense that
vantage [2–4]. Such algorithms leverage classical opti- the global minimum of the landscape may be unaffected
mizers to train the parameters in a quantum circuit, by noise [6, 23]. While some analysis has been done [44–
while employing a quantum device to efficiently estimate 46], an important open question, which has not yet been
an application-specific cost function or its gradient. By addressed, is how noise affects the asymptotic scaling of
keeping the quantum circuit depth relatively short, VQAs VQAs. More specifically, one can ask how noise affects
mitigate hardware noise and may enable near-term appli- the training process. Intuitively, incoherent noise is ex-
cations including electronic structure [5–8], dynamics [9– pected to reduce the magnitude of the gradient and hence
12], optimization [13–16], linear systems [17, 18], metrol- hinder trainability, and preliminary numerical evidence
ogy [19, 20], factoring [21], compiling [22–24], and oth- of this has been seen [47, 48], although the scaling of this
ers [25–30]. effect has not been studied.
The main open question for VQAs is their scalability to In this work, we analytically study the scaling of gra-
large problem sizes. While performing numerical heuris- dient for VQAs as a function of n, the circuit depth L,
tics for small or intermediate problem sizes is the norm and a noise parameter q. We consider a general class of
for VQAs, deriving analytical scaling results is rare for local noise models that includes depolarizing noise and
this field. Noteworthy exceptions are some recent studies certain kinds of Pauli noise. Furthermore, we investi-
of the scaling of the gradient in VQAs with the number of gate a general, abstract ansatz that allows us to encom-
qubits n [31–39]. For example, it was proven that the gra- pass many of the important ansatzes in the literature,
dient vanishes exponentially in n for randomly initialized, hence allowing us to make a general statement about
deep Hardware Efficient ansatzes [31, 32] and dissipative VQAs. This includes the Quantum Alternating Operator
quantum neural networks [33], and also for shallow depth Ansatz (QAOA) which is used for solving combinatorial
with global cost functions [34]. This vanishing gradient optimization problems [13–16] and the Unitary Coupled
phenomenon is now referred to as barren plateaus in the Cluster (UCC) Ansatz which is used in the Variational
training landscape. Fortunately, investigations into bar- Quantum Eigensolver (VQE) to solve chemistry problems
ren plateaus have spawned several promising strategies to [49–51]. This is also applicable for the Hardware Efficient
avoid them, including local cost functions [34, 40], param- Ansatz and the Hamiltonian Variational Ansatz (HVA)
eter correlation [37], pre-training [41], and layer-by-layer which are employed for various applications [52–56]. Our
training [42, 43]. Such strategies give hope that perhaps results also generalize to settings that allow for multiple2
NIBP issue. In addition, we provide numerical heuristics
that illustrate our main result for MaxCut optimization
with the QAOA, showing that NIBPs significantly im-
pact this application.
II. Results
A. General Framework
In this work we analyze a general class of parameter-
ized ansatzes U (θ) that can be expressed as a product of
L unitaries sequentially applied by layers
U (θ) = UL (θ L ) · · · U2 (θ 2 ) · U1 (θ 1 ) . (1)
FIG. 1. Schematic diagram of the Noise-Induced Bar-
ren Plateau (NIBP) phenomenon. For various appli-
Here θ = {θ l }L
l=1 is a set of vectors of continuous pa-
cations such as chemistry and optimization, increasing the rameters that are optimized to minimize a cost function
problem size often requires one to increase the depth L of the C that can be expressed as the expectation value of an
variational ansatz. We show that, in the presence of local operator O:
noise, the gradient vanishes exponentially in L and hence ex-
ponentially in the number of qubits n when L scales linearly C = Tr[OU (θ)ρU † (θ)] . (2)
in n. This can be seen in the plots on the right, which show As shown in Fig. 2, ρ is an n-qubit input state. Without
the cost function landscapes for a simple variational problem
loss of generality we assume that each Ul (θ l ) is given by
with local noise.
Y
Ul (θ l ) = e−iθlm Hlm Wlm , (3)
m
input states or training data, as in machine learning ap-
plications, often called quantum neural networks [57–61]. where Hlm are Hermitian operators, θ l = {θlm } are
Our main result (Theorem 1) is an upper bound on the continuous parameters, and Wlm denote unparametrized
magnitude of the gradient that decays exponentially with gates. We expand Hlm and O in the Pauli basis as
L, namely as 2−κ with κ = L log2 (q). Hence we find that X
i
X
the gradient vanishes exponentially in the circuit depth. Hlm = η lm · σ n = ηlm σni , O = ω · σ n = ω i σni ,
i i
Moreover, it is typical to consider L scaling as poly(n) (4)
(e.g., in the UCC Ansatz [51]), for which our main result where σni ∈ {11, X, Y, Z}⊗n are Pauli strings, and η lm and
implies an exponential decay of the gradient in n. We ω are real-valued vectors that specify the terms present
refer to this as a Noise-Induced Barren Plateau (NIBP). in the expansion. Defining Nlm = |η lm | and NO = |ω|
We remark that NIBPs can be viewed as concomitant to as the number of non-zero elements, i.e., the number of
the cost landscape concentrating around the value of the terms in the summations in Eq. (4), we say that Hlm and
cost for the maximally mixed state, and we make this O admit an efficient Pauli decomposition if Nlm , NO ∈
precise in Lemma 1. See Fig. 1 for a schematic diagram O(poly(n)), respectively.
of the NIBP phenomenon. We now briefly discuss how the QAOA, UCC, and
To be clear, any variational algorithm with a NIBP will Hardware Efficient ansatzes fit into this general frame-
have exponential scaling. In this sense, NIBPs destroy work. We refer the reader to the Methods section for
quantum speedup, as the standard goal of quantum algo- additional details. In QAOA one sequentially alternates
rithms is to avoid the typical exponential scaling of clas- the action of two unitaries as
sical algorithms. NIBPs are conceptually distinct from
the noise-free barren plateaus of Refs. [31–36]. Indeed, U (γ, β) = e−iβp HM e−iγp HP · · · e−iβ1 HM e−iγ1 HP , (5)
strategies to avoid noise-free barren plateaus [34, 37, 40–
43] do not appear to solve the NIBPs issue. where HP and HM are the so-called problem and mixer
Hamiltonian, respectively. We define NP (NM ) as the
The obvious strategy to address NIBPs is to reduce
number of terms in the Pauli decomposition of HP (HM ).
circuit complexity, or more precisely, to reduce the circuit
On the other hand, Hardware Efficient ansatzes natu-
depth. Hence, our work provides quantitative guidance
rally fit into Eqs. (1)–(3) as they are usually composed
for how small L needs to be to potentially avoid NIBPs.
of fixed gates (e.g, controlled NOTs), and parametrized
In what follows, we present our general framework fol- gates (e.g., single qubit rotations). Finally, as detailed in
lowed by our main result. We also present two extensions the Methods, the UCC ansatz can be expressed as
of our main result, one involving correlated ansatz pa- P i
Y Y i
rameters and one allowing for measurement noise. The U (θ) = Ulm (θlm ) = eiθlm i µlm σn , (6)
latter indicates that global cost functions exacerbate the lm lm3
In this case, one is primarily concerned with trainabil-
ity, and hence the gradient is a key quantity of inter-
est. These applications motivate our main result in The-
orem 1, which bounds the magnitude of the gradient. We
remark that trainability is of course also important for
VQE, and hence Theorem 1 is also of interest for this
application.
With this motivation in mind, we now present our main
results. We first present our bound on the cost function,
since one can view this as a phenomenon that naturally
accompanies our main theorem. Namely, in the following
lemma, we show that the noisy cost function concentrates
around the corresponding value for the maximally mixed
state.
FIG. 2. Setting for our analysis. An n-qubit input state
ρ is sent through a variational ansatz U (θ) composed of L Lemma 1 (Concentration of the cost function). Con-
unitary layers Ul (θ l ) sequentially acting according to Eq. (1). sider an L-layered ansatz of the form in Eq. (1). Suppose
Here, Ul denotes the quantum channel that implements the that local Pauli noise of the form of Eq. (7) with noise
unitary Ul (θ l ). The parameters in the ansatz θ = {θ l }Ll=1 are strength q acts before and after each layer as in Fig. 2.
trained to minimize a cost function that is expressed as the Then, for a cost function C e of the form in Eq. (8), the
expectation value of an operator O as in Eq. (2). We consider following bound holds
a noise model where local Pauli noise channels Nj act on each
qubit j before and after each unitary.
e − 1 Tr[O] 6 G(n) ρ − 1
C , (9)
2n 2n 1
where µilm ∈ {0, ±1}, and where θlm are the coupled where
P Nlm = |µlm | as
cluster amplitudes. Moreover, we denote G(n) = NO kωk∞ q L+1 . (10)
b
the number of non-zero elements in i µilm σni .
As shown in Fig. 2, we consider a noise model where Here k · k∞ is the infinity norm, k · k1 is the trace norm,
local Pauli noise channels Nj act on each qubit j before ω is defined in Eq. (4), and NO = |ω| is the number of
and after each unitary Ul (θ l ). The action of Nj on a non-zero elements in the Pauli decomposition of O.
local Pauli operator σ ∈ {X, Y, Z} can be expressed as
This lemma implies the cost landscape exponentially
concentrates on the value Tr[O]/2n for large n, whenever
Nj (σ) = qσ σ , (7)
the number of layers L scales linearly with the number of
where −1 < qX , qY , qZ < 1. Here, we character- qubits. While this lemma has important applications on
ize the noise strength with a single parameter q = its own, particularly for VQE, it also provides intuition
max{|qX |, |qY |, |qZ |}. Let Ul denote the channel that im- for the NIBP phenomenon, which we now state.
plements the unitary Ul (θ l ) and let N = N1 ⊗ · · · ⊗ Nn Let ∂lm C
e = ∂ C/∂θ
e lm denote the partial derivative of
denote the n-qubit noise channel. Then, the noisy cost the noisy cost function with respect to the m-th param-
function is given by eter that appears in the l-th layer of the ansatz, as in
Eq. (3). For our main technical result, we upper bound
|∂lm C|
e as a function of L and n.
e = Tr O N ◦ UL ◦ · · · ◦ N ◦ U1 ◦ N (ρ) .
C (8)
Theorem 1 (Upper bound on the partial derivative).
Consider an L-layered ansatz as defined in Eq. (1). Let
B. General Analytical Results θlm denote the trainable parameter corresponding to the
Hamiltonian Hlm in the unitary Ul (θ l ) appearing in the
There are some VQAs, such as the VQE [5] for chem- ansatz. Suppose that local Pauli noise of the form in
istry and other physical systems, where it is important Eq. (7) with noise parameter q acts before and after each
to accurately characterize the value of the cost function layer as in Fig. 2. Then the following bound holds for the
itself. We provide an important result below in Lemma 1 partial derivative of the noisy cost function
that quantitatively bounds the cost function itself, and
we envision that this bound will be especially useful in |∂lm C|
e 6 F (n), (11)
the context of VQE. On the other hand, there are other
VQAs, such as those for optimization [13–16], compil- where
√
ing [22–24], and linear systems [17, 18], where the key F (n) = 8 ln 2 NO kHlm k∞ kωk∞ n1/2 q L+1 , (12)
goal is to learn the optimal parameters and the precise
value of the cost function is either not important or can and ω is defined in Eq. (4), with number of non-zero
be computed classically after learning the parameters. elements NO .4
Let us now consider the asymptotic scaling of the func- derivative of the noisy cost with respect to θst is bounded
tion F (n) in Eq. (12). Under standard assumptions such as
as that O in Eq. (4) admits an efficient Pauli decomposi- √
e 6 8 ln 2 gNO kHlm k∞ kωk∞ n1/2 q L+1 ,
|∂st C| (16)
tion and that Hlm has bounded eigenvalues, we now state
that F (n) decays exponentially in n, if L grows linearly at all points in the cost landscape.
in n.
Remark 1 is especially important in the context of the
Corollary 1 (Noise-induced barren plateaus). Let QAOA and the UCC ansatz, as discussed below. We
i
Nlm , NO ∈ O(poly(n)) and let ηlm , ω j ∈ O(poly(n)) for note that, in the general case, a unitary of the form of
all i, j. Then the upper bound F (n) in Eq. (12) vanishes Eq. (3) cannot be implemented as a single gate on a phys-
exponentially in n as ical device. In practice one needs to compile the unitary
into a sequence of native gates. Moreover, Hamiltoni-
F (n) ∈ O(2−αn ) , (13) ans with non-commuting terms are usually approximated
with techniques such as Trotterization. This compiliation
for some positive constant α if we have
overhead potentially leads to a sequence of gates that
L ∈ Ω(n) . (14) grows with n. Remark 1 enables us to account for such
scenarios, and we elaborate on its relevance to specific
The asymptotic scaling in Eq. (13) is independent of applications in the next subsection.
l and m, i.e., the scaling is blind to the layer, or the Finally, we present an extension of our main result to
parameter within the layer, for which the derivative is the case of measurement noise. Consider a model of mea-
taken. This corollary implies that when Eq. (14) holds, surement noise where each local measurement indepen-
i.e. L grows at least linearly in n, the partial derivative dently has some bit-flip probability given by (1 − qM )/2,
|∂lm C|
e exponentially vanishes in n across the entire cost which we assume to be symmetric with respect to the 0
landscape. In other words, one observes a Noise-Induced and 1 outcomes. This leads to an additional reduction
Barren Plateau (NIBP). of our bounds on the cost function and its gradient that
We note that Eq. (14) is satisfied for all q < 1. That depends on the locality of the observable O.
is, NIBPs occur regardless of the noise strength, it only Proposition 1 (Measurement noise). Consider expand-
changes the severity of the exponential scaling. ing the observable O as a sum of Pauli strings, as in
In addition, Corollary 1 implies that NIBPs are con- Eq. (4). Let w denote the minimum weight of these
ceptually different from noise-free barren plateaus. First, strings, where the weight is defined as the number of non-
NIBPs are independent of the parameter initialization identity elements for a given string. In addition to the
strategy or the locality of the cost function. Second, noise process considered in Fig. 2, suppose there is also
NIBPs exhibit exponential decay of the gradient itself; measurement noise consisting of a tensor product of lo-
not just of the variance of the gradient, which is the cal bit-flip channels with bit-flip probability (1 − qM )/2.
hallmark of noise-free barren plateaus. Noise-free barren Then we have
plateaus allow the global minimum to sit inside deep, nar-
row valley in the landscape [34], whereas NIBPs flatten e − 1 Tr[O] 6 qM
C w 1
G(n) ρ − n (17)
the entire landscape. 2n 2 1
One of the strategies to avoid the noise-free barren
and
plateaus is to correlate parameters, i.e., to make a subset
of the parameters equal to each other [37]. We generalize w
|∂lm C|
e 6 qM F (n) (18)
Theorem 1 in the following remark to accommodate such
a setting, consequently showing that such correlated or where G(n) and F (n) are defined in Lemma 1 and The-
degenerate parameters do not help in avoiding NIBPs. orem 1, respectively.
In this setting, the result we obtain in Eq. (16) below is Proposition 1 goes beyond the noise model considered
essentially identical to that in Eq. (12) except with an in Theorem 1. It shows that in the presence of measure-
additional factor quantifying the amount of degeneracy. ment noise there is an additional contribution from the
Remark 1 (Degenerate parameters). Consider the locality of the measurement operator. It is interesting to
ansatz defined in Eqs. (1) and (3). Suppose there is a draw a parallel between Proposition 1 and noise-free bar-
subset Gst of the set {θlm } in this ansatz such that Gst ren plateaus, which have been shown to be cost-function
consists of g parameters that are degenerate: dependent and in particular depend on the locality of the
observable O [34]. The bounds in Proposition 1 similarly
depend on the locality of O. For example, when w = n,
Gst = θlm | θlm = θst . (15)
w
i.e., global observables, the factor qM will hasten the ex-
Here, θst denotes the parameter in Gst for which ponential decay. On the other hand, when w = 1, i.e., lo-
Nlm kη lm k∞ takes the largest value in the set. (θst can cal observables, the scaling is unaltered by measurement
also be thought of as a reference parameter to which all noise. In this sense, a global observable exacerbates the
other parameters are set equal in value.) Then the partial NIBP issue by making the decay more rapid with n.5
C. Application-Specific Analytical Results Moreover, weak growth of p with n combined with com-
pilation overhead could still result in an NIBP.
Here we investigate the implications of our results from Finally, we note that above we have assumed the con-
Section II B for two applications: optimization and chem- tribution of kP dominates that of kM . However, it is
istry. In particular, we derive explicit conditions for possible that for choice of more exotic mixers [16], kM
NIBPs for these applications. These conditions are de- also needs to be carefully considered to avoid NIBPs.
rived in the setting where Trotterization is used, but
Corollary 3 (Example: UCC). Let H denote a molec-
other compilation strategies incur similar asymptotic be-
ular Hamiltonian of a system of Me electrons. Consider
havior. We begin with the QAOA for optimization and
the UCC ansatz as defined in Eq. (6). If local Pauli noise
then discuss the UCC ansatz for chemistry. Finally,
of the form in Eq. (7) with noise parameter q acts before
we make a remark about the Hamiltonian Variational
and after every Ulm (θlm ) in Eq. (6), then we have
Ansatz (HVA), as well as remark that our results also ap-
ply to a generalized cost function that can employ train- √
|∂θlm C|
e 6 8 ln 2 Nblm NH kωk∞ n1/2 q L+1 , (21)
ing data.
Corollary 2 (Example: QAOA). Consider the QAOA for any coupled cluster amplitude θlm , and where O = H
with 2p trainable parameters, as defined in Eq. (5). Sup- in Eq. (2).
pose that the implementation of unitaries corresponding
to the problem Hamiltonian HP and the mixer Hamilto- Corollary 3 allows us to make general statements about
nian HM require kP - and kM -depth circuits, respectively. the trainability of UCC ansatz. We present the details for
If local Pauli noise of the form in Eq. (7) with noise pa- the standard UCC ansatz with single and double excita-
rameter q acts before and after each layer of native gates, tions from occupied to virtual orbitals [49, 68] (see Meth-
then we have ods for more details). Let Mo denote the total number of
√ spin orbitals. Then at least n = Mo qubits are required
|∂βl C|
e 6 8 ln 2 gl,P NP kHP k∞ kωk∞ n1/2 q (kP +kM )p+1 , to simulate such a system and the number of variational
(19) parameters grows as Ω(n2 Me2 ) [62, 69]. To implement the
√ 1/2 (k +k )p+1
|∂γl C|
e 6 8 ln 2 gl,M NP kHM k∞ kωk∞ n q P M , UCC ansatz on a quantum computer, the excitation op-
(20) erators are first mapped to Pauli operators using Jordan-
Wigner or Bravyi-Kitaev mappings [70, 71]. Then, using
for any choice of parameters βl , γl , and where O = HP first-order Trotterization and employing SWAP networks
in Eq. (2). Here gl,P and gl,M are the respective number [62], the UCC ansatz can be implemented in Ω(n2 Me )
of native gates parameterized by βl and γl according to depth, while assuming 1-D connectivity of qubits [62].
the compilation. Hence for the UCC ansatz, approximated by single- and
Corollary 2 follows from Remark 1 and it has interest- double-excitation operators, the upper bound in Eq. (21)
ing implications for the trainability of the QAOA. From (asymptotically) vanishes exponentially in n.
Eqs. (19) and (20), NIBPs are guaranteed if pkP scales To target strongly correlated states for molecular
linearly in n. This can manifest itself in a number of Hamiltonians, one can employ a UCC ansatz that in-
ways, which we explain below. cludes additional, generalized excitations [55, 72]. A
First, we look at the depth kP required to implement Ω(n3 ) depth circuit is required to implement the first-
one application of the problem unitary. Graph prob- order Trotterized form of this ansatz [62]. Hence NIBPs
lems containing vertices of extensive degree such as the become more prominent for generalized UCC ansatzes.
Sherrington-Kirkpatrick model inherently require Ω(n) Finally, we remark that a sparse version of the UCC
depth circuits to implement [54]. On the other hand, ansatz can be implemented in Ω(n) depth [62]. NIBPs
generic problems mapped to hardware topologies also still would occur for such ansatzes.
have the potential to incur Ω(n) depth or greater in com- Additionally, we can make the following remark about
pilation cost. For instance, implementation of MaxCut the Hamiltonian Variational Ansatz (HVA). As argued in
and k-SAT using SWAP networks on circuits with 1-D [55, 73, 74], the HVA has the potential to be an effective
connectivity requires depth Ω(n) and Ω(nk−1 ) respec- ansatz for quantum many-body problems.
tively [15, 62]. Such mappings with the aforementioned Remark 2 (Example: HVA). The HVA be thought of
compiling overhead for k > 2 are guaranteed to encounter as a generalization of the QAOA to more than two non-
NIBPs even for a fixed number of rounds p. commuting Hamiltonians. It is remarked in Ref. [56] that
Second, it appears that p values that grow at least for problems of interest the number of rounds p scales
lightly with n may be needed for quantum advantage in linearly in n. Thus, considering this growth of p and also
certain optimization problems (for example, [63–66]). In the potential growth of the compiled unitaries with n, the
addition, there are problems employing the QAOA that HVA has the potential to encounter NIBPs, by the same
explicitly require p scaling as poly(n) [21, 67]. Thus, arguments made above for the QAOA (e.g., Corollary 2).
without even considering the compilation overhead for
the problem unitary, these QAOA problems may run into Remark 3 (Quantum Machine Learning). Our results
NIBPs particularly when aiming for quantum advantage. can be extended to generalized cost functions of the form6
†
P
Ctrain = i Tr[Oi U (θ)ρi U (θ)] where {Oi } is a set of
operators each of the form (4) and {ρi } is a set of states.
This can encapsulate certain quantum machine learning
settings [57–61] that employ training data {ρi }. As an
example of an instance where NIBPs can occur, in one
study [61] an ansatz model has been proposed that requires
at least linear circuit depth in n.
D. QAOA Heuristics
To illustrate the NIBP phenomenon beyond the con-
ditions assumed in our analytical results, we numerically
implement the QAOA to solve MaxCut combinatorial op-
timization problems. We employ a realistic noise model
obtained from gate-set tomography on the IBM Ourense FIG. 3. QAOA heuristics in the presence of real-
superconducting qubit device. In the Methods section istic hardware noise: increasing number of rounds
we provide additional details on the noise model and the for fixed problem size. (a) The approximation ratio av-
optimization method employed. eraged over 100 random graphs of 5 nodes is plotted ver-
Let us first recall that a MaxCut problem is specified sus number of rounds p. The black, green, and red curves
by a graph G = (V, E) of nodes V and edges E. The respectively correspond to noise-free training, noisy train-
goal is to partition the nodes of G into two sets which ing with noise-free final cost evaluation, and noisy train-
ing with noisy final cost evaluation. The performance of
maximize the number of edges connecting nodes between
noise-free training increases with p, similar to the results in
sets. Here, the QAOA problem Hamiltonian is given by Ref. [15]. The green curve shows that the training process
1 X itself is hindered by noise, with the performance decreasing
HP = − Cij (11 − Zi Zj ) , (22) steadily with p for p > 4. The dotted blue lines correspond
2
ij∈E to known lower and upper bounds on classical performance
in polynomial time: respectively the performance guarantee
where Zi are local Pauli operators on qubit (node) i,
of the Goemans-Williamson algorithm [76] and the boundary
Cij = 1 if the nodes are connected and Cij = 0 otherwise. of known NP-hardness [77, 78]. (b) The deviation of the cost
We analyze performance in two settings. First, we fix from Tr[HP ]/2n (averaged over graphs and parameter values)
the problem size at n = 5 nodes (qubits) and vary the is plotted versus p. As p increases, this deviation decays ap-
number of rounds p (Fig. 3). Second, we fix the number proximately exponentially with p (linear on the log scale). (c)
of rounds of QAOA at p = 4 and vary the problem size The absolute value of the largest partial derivative, averaged
by increasing the number of nodes (Fig. 4). over graphs and parameter values, is plotted versus p. The
In order to quantify performance for a given n and p, partial derivatives decay approximately exponentially with p,
we randomly generate 100 graphs according to the showing evidence of Noise-Induced Barren Plateaus (NIBPs).
Erdős–Rényi model [75], such that each graph G is chosen
uniformly at random from the set of all graphs of n nodes.
For each graph we run 10 instances of the parameter opti- via noisy training. Note that evaluating the cost in a
mization, and we select the run that achieves the smallest noise-free setting has practical meaning, since the clas-
energy. At each optimization step the cost is estimated sicality of the Hamiltonian allows one to compute the
with 1000 shots. Performance is quantified by the aver- cost on a (noise-free) classical computer, after training
age approximation ratio when training the QAOA in the the parameters. For p > 4 this approximation ratio de-
presence and absence of noise. The approximation ratio creases, meaning that as p becomes larger it becomes
is defined as the lowest energy obtained via optimizing increasingly hard to find a minimizing direction to navi-
divided by the exact ground state energy of HP . gate through the cost function landscape. Moreover, the
In our first setting we observe in Fig. 3(a) that when effect of NIPBs is evident in Fig. 3(c) where we depict
training in the absence of noise, the approximation ratio the average absolute value of the largest cost function
increases with p. However, when training in the pres- partial derivative (i.e., maxlm |∂lm C|).
e This plot shows
ence of noise the performance decreases for p > 2. This an exponential decay of the partial derivative with p in
result is in accordance with Lemma 1, as the cost func- accordance with Theorem 1.
tion value concentrates around Tr[HP ]/2n as p increases. Finally, in Fig. 3(a) we contextualize our results
This concentration phenomenon can also be seen clearly with previously known two-sided bounds on classical
in Fig. 3(b), where in fact we see evidence of exponential polynomial-time performance. The lower bound cor-
decay of cost value with p. responds to the performance guarantee of the classical
In addition, we can see the effect of NIBPs as Fig. 3(a) Goemans-Williamson algorithm [76], whilst the upper
also depicts the value of the approximation ratio com- bound is at the value 16/17 which is the approximation
puted without noise by utilizing the parameters obtained ratio beyond which Max-Cut is known to be NP-hard7
scalability, and there is even less known about the im-
pact of noise on their scaling. Our work represents a
breakthrough in understanding the effect of local noise on
VQA scalability. We rigorously prove two important and
closely related phenomena: the exponential concentra-
tion of the cost function in Lemma 1 and the exponential
vanishing of the gradient in Theorem 1. We refer to the
latter as a Noise-Induced Barren Plateau (NIBP). Like
noise-free barren plateaus, NIBPs require the precision
and hence the algorithmic complexity to scale exponen-
tially with the problem size. Thus, avoiding NIBPs is
necessary for a VQA to have any hope of exponential
quantum speedup.
On the other hand, NIBPs are conceptually different
from noise-free barren plateaus [31–36]. The latter are
FIG. 4. QAOA heuristics in the presence of realistic due to random parameter initialization and hence can be
hardware noise: increasing problem size for a fixed addressed by pre-training, correlating parameters, and
number of rounds. The approximation ratio averaged over other strategies [34, 37, 40–43]. In contrast, NIBPs hold
60 random graphs of increasing number of nodes n and fixed for every point on the cost function landscape. Hence,
number of rounds p = 4 is plotted. The black, green, and red pre-training and other similar strategies do not avoid
curves respectively correspond to noise-free training, noisy NIBPs, and we explicitly demonstrate this for the pa-
training with noise-free final cost evaluation, and noisy train- rameter correlation strategy in Remark 1. At the mo-
ing with noisy final cost evaluation. (a) For a problem size of 8
ment, the only strategies we are aware of for avoiding
nodes or larger, the noisily-trained approximation ratio falls
below the performance guarantee of the classical Goemans-
NIBPs are: (1) reducing the hardware noise level, or (2)
Williamson algorithm. (b) The depth of the circuit (red improving the design of variational ansatzes such that
curve) scales linearly with the number of qubits, confirming their circuit depth scales more weakly with n. Our work
we are in a regime where we would expect to observe Noise- provides quantitative guidance for how to develop these
Induced Barren Plateaus. strategies.
An elegant feature of our work is its generality, as our
results apply to a wide range of VQAs and ansatzes.
[77, 78]. This includes the two most popular ansatzes, QAOA
In our second setting we find complementary results. for optimization and UCC for chemistry, which Corol-
In Fig. 4(a) we observe that at a problem size of 8 qubits laries 2 and 3 treat respectively. In recent times QAQA,
or larger, 4 rounds of QAOA trained on the noisy circuit UCC, and other physically motivated ansatzes have be
falls short of the performance guarantees of the classi- touted as the potential solution to trainability issues due
cal Goemans-Williamson algorithm. As we increase the to (noise-free) barren plateaus, while Hardware Efficient
number of qubits, we also observe this increases the depth ansatzes, which minimize circuit depth, have been re-
of the circuit linearly (Fig. 4(b)), thus confirming we are garded as problematic. Our work swings the pendulum in
in a regime of NIBPs. the other direction: any additional circuit depth that an
Our numerical results show that training the QAOA ansatz incorporates (regardless of whether it is physically
in the presence of a realistic noise model significantly motivated) will hurt trainability and potentially lead to
affects the performance. The concentration of cost and a NIBP. This suggests that Hardware Efficient ansatzes
the NIBP phenomenon are both also clearly visible in our are in fact worth exploring further, provided one has an
data. Even though we observe performance for n = 5 appropriate strategy to avoid noise-free barren plateaus.
and p = 4 that is NP-hard to achieve classically, any This claim is supported by recent state-of-the-art imple-
possible advantage would be lost for large problem sizes mentations for optimization [54] and chemistry [53] using
or circuit depth due to bad scaling. Hence, noise appears such ansatzes.
to be a crucial factor to account for when attempting to We believe our work has particular relevance to opti-
understand the performance of QAOA. mization. For combinatorial optimization problems, such
as MaxCut on 3-regular graphs, the compilation of a sin-
gle instance of the problem unitary e−iγHP can require an
III. Discussion Ω(n)-depth circuit [54]. Therefore, for a constant number
of rounds p of the QAOA, the circuit depth grows at least
The success of NISQ computing largely depends on the linearly with n. From Theorem 1, it follows that NIBPs
scalability of Variational Quantum Algorithms (VQAs), can occur for practical QAOA problems, even for con-
which are widely viewed as the best hope for near-term stant number of rounds. Furthermore, even neglecting
quantum advantage for various applications. Only a the aforementioned linear compilation overhead, NIBPs
small handful of works have analytically studied VQA are guaranteed (asymptotically) if p grows in n. Such8
growth has been shown to be necessary in certain in-
stances of MaxCut [63] as well as for other optimization
problems [21, 67], and hence NIBPs are especially rele-
vant in these cases.
While it is well known that decoherence ultimately lim-
its the depth of quantum circuits in the NISQ era, there
was an interesting open question (prior to our work) as
to whether one could still train the parameters of a vari-
ational ansatz in the high decoherence limit. This ques-
tion was especially important for VQAs for optimization,
compiling, and linear systems, which are applications
that do not require accurate estimation of cost func-
tions on the quantum computer. Our work essentially
provides a negative answer to this question. Naturally,
important future work will involve extending our results
to more general (e.g., non-unital) noise models, and nu-
merically testing the tightness of our bounds. Moreover, FIG. 5. Special cases of our general ansatz. (a) QAOA
our work emphasizes the importance of short-depth vari- problem unitary e−iγHP for the ring-of-disagrees MaxCut
problem, with Hamiltonian HP = 21 j Zj Zj+1 . (b) Hard-
P
ational ansatzes. Hence a crucial research direction for
the success of VQAs will be the development of methods ware Efficient ansatz composed of CNOTs and single qubit
to reduce ansatz depth. rotations around the y-axis Ry (θ). (c) Unitary for the expo-
nential e−iθY1 Z2 Z3 X4 . This type of circuit is a representative
component of the UCC ansatz.
IV. Methods
2. Hardware Efficient Ansatz
A. Special Cases of Our Ansatz
The goal of the Hardware Efficient ansatz is to reduce
Here we discuss how the the QAOA, the Hardware the gate overhead (and hence the circuit depth) which
Efficient ansatz, and the UCC ansatz fit into the general arises when implementing a general unitary as in (3).
framework of Section II A. Hence, when employing a specific quantum hardware the
parametrized gates e−iθlm Hlm and the unparametrized
gates Wlm are taken from a gate alphabet composed of
native gates to that hardware. Figure 5(b) shows an
1. Quantum Alternating Operator Ansatz example of a Hardware Efficient ansatz where the gate
alphabet is composed of rotations around the y axis and
of CNOTs.
The QAOA can be understood as a discretized adi-
abatic transformation where the goal is to prepare the
ground state of a given Hamiltonian HP . The order p
of the Trotterization determines the solution precision
3. Unitary Coupled Cluster Ansatz
and the circuit depth. Given an initial state |si, usually
the linear superposition of all elements of the computa-
tional basis |si = |+i⊗n , the ansatz corresponds to the This ansatz is employed to estimate the ground state
sequential application of two unitaries UP (γl ) = e−iγl HP energy of the molecular Hamiltonian. In the second
and UM (βl ) = e−iβl HM . These alternating unitaries are quantization, and within the Born-Oppenheimer approx-
usually known as the problem and mixer unitary, respec- imation, the molecular Hamiltonian of aPsystem of Me
†
tively. Here γ = {γk }L L
l=1 and β = {βk }l=1 are vectors electrons can be expressed as: H = pq hpq ap aq +
of variational parameters which determine how long each 1 † † †
P
2 pqrs hpqrs ap aq ar as , where {ap } ({aq }) are Fermionic
unitary is applied and which must be optimized to mini- creation (annihilation) operators. Here, hpq and hpqrs
mize the cost function C, defined as the expectation value respectively correspond to the so-called one- and two-
electron integrals [49, 68]. The ground state energy of
C = hγ, β|HP |γ, βi = Tr[HP |γ, βihγ, β|] , (23) H can be estimated with the VQE algorithm by prepar-
ing a reference state, normally taken to be the Hartree-
where |γ, βi = U (γ, β)|si is the QAOA variational state, Fock (HF) mean-field state |ψ0 i, and acting on it with a
and where U (γ, β) is given by (5). In Fig. 5(a) we depict parametrized UCC ansatz.
the circuit description of a QAOA ansatz for a specific The action of a UCC ansatz with single (T1 ) and double
Hamiltonian where kP = 6. (T2 ) excitations is given by |ψi = exp(T − T † )|ψ0 i, where9
T = T1 + T2 , and where Let us now present two lemmas that reflect these
two parts of the proof. The action of the noise in
ti,j a†a a†b aj ai .
X X a,b
T1 = tai a†a ai , T2 = (24) (7) on the operator Λ is to map the elements of λ as
i∈occ i,j∈occ N x(i) y(i) z(i)
a∈vir a,b∈vir λi −−→ λ0i = qX qY qZ λi where x(i), y(i), and z(i)
respectively denote the number of X, Y , and Z oper-
Here the i and j indices range over “occupied” orbitals ators in the i-th Pauli string. Recall the definition
whereas the a and b indices range over “virtual” or- q = max{|qX |, |qY |, |qZ |}. Since x(i) + y(i) + z(i) > 1,
bitals [49, 68]. The coefficients tai and ta,b i,j are called the inequality |λ0 | 6 q|λ| always holds. We use this re-
coupled cluster amplitudes. For simplicity, we denote lationship, along with Weyl’s inequality and the unitary
these amplitudes {tai , ta,b
i,j } as {θlm }. Similarly, by denot-
invariance of Schatten norms to show that for an operator
ing the excitation operators {a†a ai , a†a a†b aj ai } as {τlm }, of the form (25) we have
the UCC P ansatz can be written in a compact form as W k (Λ) ∞
6 λ0 + q k λ 1
(26)
†
U (θ) = e lm θlm (τlm −τlm ) . In order to implement U (θ)
one maps the fermionic operators to spin operators by where W k is a channel composed of k unitaries inter-
means of the Jordan-Wigner or the Bravyi-Kitaev trans- leaved with noise channels of the form (7). The second
† lemma we present is a bound on the relative entropy by
P i i [70, 71], which allows us to write (τlm −τlm ) =
formations
Müller-Hermes et al. [79] which states that
i i µlm σn . Then, from a first-order Trotterization we
obtain (6). Here, µilm ∈ {0, ±1}. In Fig. 5(c) we depict D W(ρ) 11⊗n /2n 6 q 2k D ρ 11⊗n /2n
(27)
the circuit description of a representative component of
where we recall that D ρ 11⊗n /2n itself is always upper
the UCC ansatz.
bounded by n for any n-qubit quantum state ρ.
Now that we have the main tools we present a sketch
B. Proof of Theorem 1 of the proof. In order to analyze the partial derivative of
the cost function ∂lm C
e = Tr [O ∂lm ρL ] we first note that
Here we outline the proof for our main result on Noise- the output state ρL can be expressed as
Induced Barren Plateaus. We refer the reader to the Sup- ρL = (Wa ◦ Wb ) (ρ0 ) = Wa (ρ̄l ) , (28)
plementary Information for additional details. We note
that Lemma 1 and Remark 1 follow from similar steps where ρ0 is the input state and
and their proofs are also detailed in the Supplementary +
Wa = N ◦ UL ◦ · · · ◦ Ul+1 ◦ N ◦ Ulm , (29)
Information. Moreover, we remark that Corollaries 1, 2, −
Wb = Ulm ◦ N ◦ Ul−1 ◦ · · · ◦ N ◦ U1 ◦ N , (30)
and 3 follow in a straightforward manner from a direct
application of Theorem 1 and Remark 1. ±
where Ulm are channels that implement the unitaries
Throughout our calculations we find it useful to use −
Ulm = s6m e−iθls Hls and Ulm
Q +
= s>m e−iθls Hls such
Q
the expansion of operators in the Pauli tensor product + −
that Ul = Ulm · Ulm . For simplicity of notation here we
basis. Given an n-qubit Hermitian operator Λ, one can
have omitted the parameter dependence on the concate-
always consider the decomposition
nation of channels. Additionally, we have introduced the
Λ = λ0 11⊗n + λ · σ n , (25) notation ρ̄l = Wb (ρ0 ) and it is straightforward to show
n
that
where λ0 ∈ R and λ ∈ R4 −1 . Note that here we redefine
∂lm ρ̄l = −i[Hlm , ρ̄l ] . (31)
the vector of Pauli strings σ n as a vector of length 4n − 1
which excludes 11⊗n . Using the tracial matrix Hölder’s inequality [80], we
Central to our proof is to understand how operators can write
are mapped by concatenations of unitary transformations e = Tr Wa† (O) ∂lm ρ̄l
∂lm C (32)
and noise channels. We do this through two lenses. First,
†
given an operator Λ we investigate how various `p -norms 6 Wa (O) ∞ ∂lm ρ̄l 1 , (33)
of λ are related at different points in the evolution. Such
quantities are well suited to study in our setting as we where Wa† is the adjoint map of Wa . The two terms
can use the transfer matrix formalism in the Pauli ba- in the product can then be bounded with the above
sis, that is, to represent a channel N with the matrix two techniques. Using (26) we find Wa† (O) ∞ 6
(TN )ij = 21n Tr σni N (σnj ) . Indeed, we see that the noise q L−l+1 NO ω ∞ for the first term. We bound the sec-
model in (7) has a diagonal Pauli transfer matrix, which ond term by using (31), a bound on Schatten norms
motivates this choice of attack. The second quantity we of commutators [81], quantum
Pinsker’s
√ inequality [82],
use is the relative entropy D ρ 11⊗n /2n between a state and (27) to obtain ∂lm ρ̄l 1 6 8 ln 2 Hlm ∞ n1/2 q l .
ρ and the maximally mixed state. This is also useful Putting the two parts together we obtain
to study due to the strong data processing inequality in √
e 6 8 ln 2 NO kHlm k∞ kωk∞ n1/2 q L+1 ,
∂lm C (34)
Ref. [79] which quantifies how noise maps ρ closer to the
maximally mixed state. completing the proof.10
w
C. Proof of Proposition 1 with qM for each term in the sum. This gives an ex-
tra locality-dependent factor in the bound on the partial
Here we sketch the proof of Proposition 1, with addi- derivative:
tional details being presented in the Supplementary In- w
|∂lm C|
e 6 qM F (n). (40)
formation.
We model measurement noise as a tensor product of An analogous reasoning leads to the following result
independent local classical bit-flip channels, which math- for the concentration of the cost function:
ematically corresponds to modifying the local POVM el-
ements P0 = |0ih0| and P1 = |1ih1| as follows: e − 1 Tr O 6 q w G(n).
C (41)
M
2n
1 + qM 1 − qM
P0 = |0ih0| → Pe0 = |0ih0| + |1ih1| (35)
2 2
1 − qM 1 + qM D. Details of Numerical Implementations
P1 = |1ih1| → Pe1 = |0ih0| + |1ih1| . (36)
2 2
The noise model employed in our numerical simula-
In turn, it follows that one can also model this measure- tions was obtained by performing one- and two-qubit
ment noise as a tensor product of local depolarizing chan- gate-set tomography [83, 84] on the five-qubit IBM Q
nels with depolarizing probability 1 > (1 − qM )/2 > 0, Ourense superconducting qubit device. The process ma-
which we indicate by NM . The channel is applied di- trices for each gate native to the device’s alphabet, and
the state preparation and measurement noise are de-
P i to thei measurement operator such that NM (O) =
rectly
i ω NM (σn ) = ω e · σ n . Here ω
e is a vector of coeffi- scribed in Ref. [85, Apendix B]. In addition, the opti-
cients ω
w(i)
e i = qM ω i , where w(i) = x(i) + y(i) + z(i) is mization for the MaxCut problems was performed using
the weight of the Pauli string. Here we recall that we an optimizer based on the Nelder-Mead simplex method.
have respectively defined x(i), y(i), z(i) as the number
of Pauli operators X, Y , and Z in the i-th Pauli string.
V. ACKNOWLEDGEMENTS
Let us first focus on the partial derivative of the cost.
In the presence of measurement noise we then have
We thank Daniel Stilck França for helpful discussions
and for pointing us to Ref. [79]. Research presented in
e = 1 Tr (e
h i
∂lm C ω · σ n )(g (L) · σ n ) (37) this article was supported by the Laboratory Directed
2n Research and Development program of Los Alamos Na-
=ωe · g (L) . (38) tional Laboratory under project number 20190065DR.
SW and EF acknowledge support from the U.S. Depart-
Which means that |∂lm C|
e = |eω · g (L) |. We then examine ment of Energy (DOE) through a quantum computing
the inner product in an element-wise fashion: program sponsored by the LANL Information Science
& Technology Institute. MC and AS were also sup-
X (L)
X w(i) (L) ported by the Center for Nonlinear Studies at LANL.
ω · g (L) | 6
|e |e
ωi ||gi | 6 qM |ωi ||gi | . (39) PJC also acknowledges support from the LANL ASC Be-
i i yond Moore’s Law project. LC and PJC were also sup-
ported by the U.S. Department of Energy (DOE), Of-
Therefore, defining w = mini w(i) as the minimum fice of Science, Office of Advanced Scientific Computing
weight of the Pauli strings in the decomposition of O, Research, under the Quantum Computing Applications
w(i) w w(i)
we have that qM 6 qM , and hence we can replace qM Team (QCAT) program.
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