Learning to predict arbitrary quantum processes Hsin-Yuan Huang1Sitan Chen2and John Preskill1 3 1Institute for Quantum Information and Matter and

2025-05-02 0 0 2.52MB 52 页 10玖币
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Learning to predict arbitrary quantum processes
Hsin-Yuan Huang,1Sitan Chen,2and John Preskill1, 3
1Institute for Quantum Information and Matter and
Department of Computing and Mathematical Sciences, Caltech, Pasadena, CA, USA
2Department of Electrical Engineering and Computer Sciences, UC Berkeley, Berkeley, CA, USA
3AWS Center for Quantum Computing, Pasadena, CA, USA
(Dated: April 18, 2023)
We present an efficient machine learning (ML) algorithm for predicting any unknown quantum
process ℰover 𝑛qubits. For a wide range of distributions 𝒟on arbitrary 𝑛-qubit states, we show that
this ML algorithm can learn to predict any local property of the output from the unknown process ℰ,
with a small average error over input states drawn from 𝒟. The ML algorithm is computationally
efficient even when the unknown process is a quantum circuit with exponentially many gates. Our
algorithm combines efficient procedures for learning properties of an unknown state and for learning
a low-degree approximation to an unknown observable. The analysis hinges on proving new norm
inequalities, including a quantum analogue of the classical Bohnenblust-Hille inequality, which we
derive by giving an improved algorithm for optimizing local Hamiltonians. Numerical experiments
on predicting quantum dynamics with evolution time up to 106and system size up to 50 qubits
corroborate our proof. Overall, our results highlight the potential for ML models to predict the
output of complex quantum dynamics much faster than the time needed to run the process itself.
I. INTRODUCTION
Learning complex quantum dynamics is a fundamental problem at the intersection of machine learning
(ML) and quantum physics. Given an unknown 𝑛-qubit completely positive trace preserving (CPTP) map ℰ
that represents a physical process happening in nature or in a laboratory, we consider the task of learning to
predict functions of the form
𝑓(𝜌, 𝑂) = tr(𝑂ℰ(𝜌)),(1)
where 𝜌is an 𝑛-qubit state and 𝑂is an 𝑛-qubit observable. Related problems arise in many fields of research,
including quantum machine learning [1–10], variational quantum algorithms [11–17], machine learning for
quantum physics [18–29], and quantum benchmarking [30–36]. As an example, for predicting outcomes of
quantum experiments [8,37,38], we consider 𝜌to be parameterized by a classical input 𝑥,ℰis an unknown
process happening in the lab, and 𝑂is an observable measured at the end of the experiment. Another example
is when we want to use a quantum ML algorithm to learn a model of a complex quantum evolution with the
hope that the learned model can be faster [7,11,12].
As an 𝑛-qubit CPTP map ℰconsists of exponentially many parameters, prior works, including those based
on covering number bounds [4,7,8,37], classical shadow tomography [33,39], or quantum process tomography
[30–32], require an exponential number of data samples to guarantee a small constant error for predicting
outcomes of an arbitrary evolution ℰunder a general input state 𝜌. To improve upon this, recent works
[4,7,8,37,40] have considered quantum processes ℰthat can be generated in polynomial-time and shown
that a polynomial amount of data samples suffices to learn tr(𝑂ℰ(𝜌)) in this restricted class. However, these
results still require exponential computation time.
In this work, we present a computationally-efficient ML algorithm that can learn a model of an arbitrary
unknown 𝑛-qubit process ℰ, such that given 𝜌sampled from a wide range of distributions over arbitrary 𝑛-qubit
states and any 𝑂in a large physically-relevant class of observables, the ML algorithm can accurately predict
𝑓(𝜌, 𝑂) = tr(𝑂ℰ(𝜌)). The ML model can predict outcomes for highly entangled states 𝜌after learning from
a training set that only contains data for random product input states and randomized Pauli measurements
on the corresponding output states. The training and prediction of the proposed ML model are both efficient
even if the unknown process ℰis a Hamiltonian evolution over an exponentially long time, a quantum circuit
with exponentially many gates, or a quantum process arising from contact with an infinitely large environment
for an arbitrarily long time. Furthermore, given few-body reduced density matrices (RDMs) of the input state
𝜌, the ML algorithm uses only classical computation to predict output properties tr(𝑂ℰ(𝜌)).
The proposed ML model is a combination of efficient ML algorithms for two learning problems: (1) predict-
ing tr(𝑂𝜌)given a known observable 𝑂and an unknown state 𝜌, and (2) predicting tr(𝑂𝜌)given an unknown
observable 𝑂and a known state 𝜌. We give sample- and computationally-efficient learning algorithms for
both problems. Then we show how to combine the two learning algorithms to address the problem of learn-
ing to predict tr(𝑂ℰ(𝜌)) for an arbitrary unknown 𝑛-qubit quantum process ℰ. Together, the sample and
computational efficiency of the two learning algorithms implies the efficiency of the combined ML algorithm.
In order to establish the rigorous guarantee for the proposed ML algorithms, we consider a different task:
optimizing a 𝑘-local Hamiltonian 𝐻=𝑃∈{𝐼,𝑋,𝑌,𝑍}⊗𝑛𝛼𝑃𝑃. We present an improved approximate opti-
arXiv:2210.14894v3 [quant-ph] 15 Apr 2023
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A high-complexity quantum process
Learning …
Properties of
the input state
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A low-complexity
learned model
Properties of
the output state
Some Repetitions
A Classical
Dataset
Classical machine
A Learned
Model
Figure 1: Learning to predict an arbitrary unknown quantum process ℰ.Given an unknown quantum process ℰ
with arbitrarily high complexity, and a classical dataset obtained from evolving random product states under ℰand
performing randomized Pauli measurements on the output states. We give an algorithm that can learn a low-complexity
model for predicting the local properties of the output states given the local properties of the input states.
mization algorithm that finds either a maximizing/minimizing state |𝜓⟩with a rigorous lower/upper bound
guarantee on the energy ⟨𝜓|𝐻|𝜓⟩in terms of the Pauli coefficients 𝛼𝑃of 𝐻. The rigorous bounds improve
upon existing results on optimizing 𝑘-local Hamiltonians [41–44]. We then use the improved optimization
algorithm to give a constructive proof of several useful norm inequalities relating the spectral norm ‖𝑂‖of
an observable 𝑂and the ℓ𝑝-norm of the Pauli coefficients 𝛼𝑃associated to the observable 𝑂. The proof
resolves a recent conjecture in [45] about the existence of quantum Bohnenblust-Hille inequalities. These
norm inequalities are then used to establish the efficiency of the proposed ML algorithms.
II. LEARNING QUANTUM STATES, OBSERVABLES, AND PROCESSES
Before proceeding to state our main results in greater detail, we describe informally the learning tasks
discussed in this paper: what do we mean by learning a quantum state, observable, and process?
A. Learning an unknown state
It is possible in principle to provide a complete classical description of an 𝑛-qubit quantum state 𝜌. However
this would require an exponential number of experiments, which is not at all practical. Therefore, we set a
more modest goal: to learn enough about 𝜌to predict many of its physically relevant properties. We specify
a family of target observables {𝑂𝑖}and a small target accuracy 𝜖. The learning procedure is judged to be
successful if we can predict the expectation value tr(𝑂𝑖𝜌)of every observable in the family with error no larger
than 𝜖.
Suppose that 𝜌is an arbitrary and unknown 𝑛-qubit quantum state, and suppose that we have access
to 𝑁identical copies of 𝜌. We acquire information about 𝜌by measuring these copies. In principle, we
could consider performing collective measurements across many copies at once. Or we might perform single-
copy measurements sequentially and adaptively; that is, the choice of measurement performed on copy 𝑗
could depend on the outcomes obtained in measurements on copies 1,2,3,...𝑗−1.The target observables we
consider are bounded-degree observables. A bounded-degree 𝑛-qubit observable 𝑂is a sum of local observables
(each with support on a constant number of qubits independent of 𝑛) such that only a constant number
(independent of 𝑛) of terms in the sum act on each qubit. Most thermodynamic quantities that arise in
quantum many-body physics can be written as a bounded-degree observable 𝑂, such as a geometrically-local
Hamiltonian or the average magnetization.
In the learning protocols discussed in this paper, the measurements are neither collective nor adaptive.
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Instead, we fix an ensemble of possible single-copy measurements, and for each copy of 𝜌we independently
sample from this ensemble and perform the selected measurement on that copy. Thus there are two sources of
randomness in the protocol — the randomly chosen measurement on each copy, and the intrinsic randomness
of the quantum measurement outcomes. If we are unlucky, the chosen measurements and/or the measurement
outcomes might not be sufficiently informative to allow accurate predictions. We will settle for a protocol
that achieves the desired prediction task with a high success probability.
For the protocol to be practical, it is highly advantageous for the sampled measurements to be easy to
perform in the laboratory, and easy to describe in classical language. The measurements we consider, random
Pauli measurements, meet both of these criteria. For each copy of 𝜌and for each of the 𝑛qubits, we choose
uniformly at random to measure one of the three single-qubit Pauli observables 𝑋,𝑌, or 𝑍. This learning
method, called classical shadow tomography, was analyzed in [46], where an upper bound on the sample
complexity (the number 𝑁of copies of 𝜌needed to achieve the task) was expressed in terms of a quantity
called the shadow norm of the target observables.
In this work, using a new norm inequality derived here, we improve on the result in [46] by obtaining a
tighter upper bound on the shadow norm for bounded degree observables. The upshot is that, for a fixed
target accuracy 𝜖, we can predict all bounded-degree observables with spectral norm less than 𝐵by performing
random Pauli measurement on
𝑁=𝒪log(𝑛)𝐵2/𝜖2(2)
copies of 𝜌. This result improves upon the previously known bound of 𝒪(𝑛log(𝑛)𝐵2/𝜖2). Furthermore, we
derive a matching lower bound on the number of copies required for this task, which applies even if collective
measurements across many copies are allowed.
B. Learning an unknown observable
Now suppose that 𝑂is an arbitrary and unknown 𝑛-qubit observable. We also consider a distribution 𝒟
on 𝑛-qubit quantum states. This distribution, too, need not be known, and it may include highly entangled
states. Our goal is to find a function ℎ(𝜌)which predicts the expectation value tr(𝑂𝜌)of the observable 𝑂
on the state 𝜌with a small mean squared error:
E
𝜌∼𝒟 |ℎ(𝜌)−tr(𝑂𝜌)|2≤𝜖.
To define this learning task, it is convenient to assume that we can access training data of the form
{𝜌ℓ,tr (𝑂𝜌ℓ)}𝑁
ℓ=1 ,(3)
where 𝜌ℓis sampled from the distribution 𝒟. In practice, though, we cannot directly access the exact value
of the expectation value tr (𝑂𝜌ℓ); instead, we might measure 𝑂multiple times in the state 𝜌ℓto obtain an
accurate estimate of the expectation value. Furthermore, we don’t necessarily need to sample states from 𝒟
to achieve the task. We might prefer to learn about 𝑂by accessing its expectation value in states drawn from
a different ensemble.
A crucial idea of this work is that we can learn 𝑂efficiently if the distribution 𝒟has suitably nice features.
Specifically, we consider distributions that are invariant under single-qubit Clifford gates applied to any one
of the 𝑛qubits. We say that such distributions are locally flat, meaning that the probability weight assigned
to an 𝑛-qubit state is unmodified (i.e., the distribution appears flat) when we locally rotate any one of the
qubits.
An arbitrary observable 𝑂can be expanded in terms of the Pauli operator basis:
𝑂=
𝑃∈{𝐼,𝑋,𝑌,𝑍}⊗𝑛
𝛼𝑃𝑃. (4)
Though there are 4𝑛Pauli operators, if the distribution 𝒟is locally flat and 𝑂has a constant spectral norm,
we can approximate the sum over 𝑃by a truncated sum
𝑂(𝑘)=
𝑃∈{𝐼,𝑋,𝑌,𝑍}⊗𝑛:|𝑃|≤𝑘
𝛼𝑃𝑃. (5)
including only the Pauli operators 𝑃with weight |𝑃|up to 𝑘, those acting nontrivially on no more than 𝑘
qubits. The mean squared error incurred by this truncation decays exponentially with 𝑘. Therefore, to learn
𝑂with mean squared error 𝜖it suffices to learn this truncated approximation to 𝑂, where 𝑘=𝒪(log(1/𝜖)).
Furthermore, using norm inequalities derived in this paper, we show that for the purpose of predicting the
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expectation value of this truncated operator it suffices to learn only a few relatively large coefficients 𝛼𝑃,
while setting the rest to zero. The upshot is that, for a fixed target error 𝜖, an observable with constant
spectral norm can be learned from training data with size 𝒪(log 𝑛), where the classical computational cost of
training and predicting is 𝑛𝒪(𝑘).
Usually, in machine learning, after learning from a training set sampled from a distribution 𝒟, we can
only predict new instances sampled from the same distribution 𝒟. We find, though, that for the purpose
of learning an unknown observable, there is a particular locally flat distribution 𝒟′such that learning to
predict under 𝒟′suffices for predicting under any other locally flat distribution. Namely, we samples from the
𝑛-qubit state distribution 𝒟′by preparing each one of the 𝑛qubits in one of the six Pauli operator eigenstates
{|0⟩,|1⟩,|+⟩,|−⟩,|𝑦+⟩,|𝑦−⟩}, chosen uniformly at random. Pleasingly, preparing samples from 𝒟′is not
only sufficient for our task, but also easy to do with existing quantum devices.
After training is completed, to predict tr(𝑂𝜌)for a new state 𝜌drawn from the distribution 𝒟, we need
to know some information about 𝜌. The state 𝜌, like the operator 𝑂, can be expanded in terms of Pauli
operators, and when we replace 𝑂by its weight-𝑘truncation, only the truncated part of 𝜌contributes to
its expectation value. Thus if the 𝑘-body reduced density matrices (RDMs) for states drawn from 𝒟are
known classically, then the predictions can be computed classically. If the states drawn from 𝒟are presented
as unknown quantum states, then we can learn these 𝑘-body RDMs efficiently (for small 𝑘) using classical
shadow tomography, and then proceed with the classical computation to obtain a predicted value of tr(𝑂𝜌).
C. Learning an unknown process
Now suppose that ℰis an arbitrary and unknown quantum process mapping 𝑛qubits to 𝑛qubits. Let
{𝑂𝑖}be a family of target observables, and 𝒟be a distribution on quantum states. We assume the ability
to repeatedly access ℰfor a total of 𝑁times. Each time we can apply ℰto an input state of our choice, and
perform the measurement of our choice on the resulting output. In principle we could allow input states that
are entangled across the 𝑁channel uses, and allow collective measurements across the 𝑁channel outputs.
But here we confine our attention to the case where the 𝑁inputs are unentangled, and the channel outputs
are measured individually. Our goal is to find a function ℎ(𝜌, 𝑂)which predicts, with a small mean squared
error, the expectation value of 𝑂𝑖in the output state ℰ(𝜌)for every observable 𝑂𝑖in the family {𝑂𝑖}:
E
𝜌∼𝒟 |ℎ(𝜌, 𝑂𝑖)−tr(𝑂𝑖ℰ(𝜌))|2≤𝜖. (6)
Our main result is that this task can be achieved efficiently if 𝑂𝑖is a bounded-degree observable and 𝒟is
locally flat. That is, 𝑁, the number of times we access ℰ, and the computational complexity of training and
prediction, scale reasonably with the system size 𝑛and the target accuracy 𝜖.
To prove this result, we observe that the task of learning an unknown quantum process can be reduced to
learning unknown states and learning unknown observables. If 𝜌ℓis sampled from the distribution 𝒟, then,
since ℰis unknown, ℰ(𝜌ℓ)should be regarded as an unknown quantum state. Suppose we learn this state;
that is, after preparing and measuring ℰ(𝜌ℓ)sufficiently many times we can accurately predict the expectation
value tr(𝑂𝑖ℰ(𝜌ℓ)) for each target observable 𝑂𝑖.
Now notice that tr(𝑂𝑖ℰ(𝜌ℓ)) = tr(ℰ†(𝑂𝑖)𝜌ℓ), where ℰ†is the (Heisenberg-picture) map dual to ℰ. Since ℰ†
is unknown, ℰ†(𝑂𝑖)should be regarded as an unknown observable. Suppose we learn this observable; that is,
using the dataset {𝜌ℓ,tr(ℰ†(𝑂𝑖)𝜌ℓ)}as training data, we can predict tr(ℰ†(𝑂𝑖)𝜌)for 𝜌drawn from 𝒟with a
small mean squared error. This achieves the task of learning the process ℰfor state distribution 𝒟and target
observable 𝑂𝑖.
Having already shown that arbitrary quantum states can be learned efficiently for the purpose of predicting
expectation values of bounded-degree observables, and that arbitrary observables can be learned efficiently
for locally flat input state distributions, we obtain our main result. Since the distribution 𝒟is locally flat,
it suffices to learn the low-degree truncated approximation to the unknown operator ℰ†(𝑂𝑖), incurring only a
small mean squared error. To predict tr(ℰ†(𝑂𝑖)𝜌), then, it suffices to know only the few-body RDMs of the
input state 𝜌. For any input state 𝜌, these few-body density matrices can be learned efficiently using classical
shadow tomography.
As noted above in the discussion of learning observables, the states 𝜌ℓin the training data need not
be sampled from 𝒟. To learn a low-degree approximation to ℰ†(𝑂𝑖), it suffices to sample from a locally
flat distribution on product states. Even if we sample only product states during training, we can make
accurate predictions for highly entangled input states. We also emphasize again that the unknown process
ℰis arbitrary. Even if ℰhas quantum computational complexity exponential in 𝑛, we can learn to predict
tr(𝑂ℰ(𝜌)) accurately and efficiently, for bounded-degree observables 𝑂and for any locally flat distribution on
the input state 𝜌.
5
III. ALGORITHM FOR LEARNING AN UNKNOWN QUANTUM PROCESS
Consider an unknown 𝑛-qubit quantum process ℰ(a CPTP map). Suppose we have obtained a classical
dataset by performing 𝑁randomized experiments on ℰ. Each experiment prepares a random product state
|𝜓(in)⟩=𝑛
𝑖=1 |𝑠(in)
𝑖⟩, passes through ℰ, and performs a randomized Pauli measurement [46,47] on the output
state. Recall that a randomized Pauli measurement measures each qubit of a state in a random Pauli basis
(𝑋,𝑌or 𝑍) and produces a measurement outcome of |𝜓(out)⟩=𝑛
𝑖=1 |𝑠(out)
𝑖⟩, where |𝑠(out)
𝑖⟩ ∈ stab1,
{|0⟩,|1⟩,|+⟩,|−⟩,|𝑦+⟩,|𝑦−⟩}. We denote the classical dataset of size 𝑁to be
𝑆𝑁(ℰ),|𝜓(in)
ℓ⟩=
𝑛
𝑖=1 |𝑠(in)
ℓ,𝑖 ⟩,|𝜓(out)
ℓ⟩=
𝑛
𝑖=1 |𝑠(out)
ℓ,𝑖 ⟩𝑁
ℓ=1
,(7)
where |𝑠(in)
ℓ,𝑖 ⟩,|𝑠(out)
ℓ,𝑖 ⟩ ∈ stab1. Each product state is represented classically with 𝒪(𝑛)bits. Hence, the classical
dataset 𝑆𝑁(ℰ)is of size 𝒪(𝑛𝑁)bits. The classical dataset can be seen as one way to generalize the notion of
classical shadows of quantum states [46] to quantum processes. Our goal is to design an ML algorithm that
can learn an approximate model of ℰfrom the classical dataset 𝑆𝑁(ℰ), such that for a wide range of states 𝜌
and observables 𝑂, the ML model can predict a real value ℎ(𝜌, 𝑂)that is approximately equal to tr(𝑂ℰ(𝜌)).
A. ML algorithm
We are now ready to state the proposed ML algorithm. At a high level, the ML algorithm learns a low-
degree approximation to the unknown 𝑛-qubit CPTP map ℰ. Despite the simplicity of the ML algorithm,
several ideas go into the design of the ML algorithm and the proof of the rigorous performance guarantee.
These ideas are presented in Section IV.
Let 𝑂be an observable with ‖𝑂‖ ≤ 1that is written as a sum of few-body observables, where each qubit is
acted by 𝒪(1) of the few-body observables. We denote the Pauli representation of 𝑂as 𝑄∈{𝐼,𝑋,𝑌,𝑍}⊗𝑛𝑎𝑄𝑄.
By definition of 𝑂, there are 𝒪(𝑛)nonzero Pauli coefficients 𝑎𝑄. We consider a hyperparameter ˜𝜖 > 0;
roughly speaking ˜𝜖will scale inverse polynomially in the dataset size 𝑁from Eq. (12). For every Pauli
observable 𝑃∈ {𝐼, 𝑋, 𝑌, 𝑍}⊗𝑛with |𝑃| ≤ 𝑘= Θ(log(1/𝜖)), the algorithm computes an empirical estimate for
the corresponding Pauli coefficient 𝛼𝑃via
^𝑥𝑃(𝑂) = 1
𝑁
𝑁
ℓ=1
tr 𝑃
𝑛
𝑖=1 |𝑠(in)
ℓ,𝑖 ⟩⟨𝑠(in)
ℓ,𝑖 |tr 𝑂
𝑛
𝑖=1 3|𝑠(out)
ℓ,𝑖 ⟩⟨𝑠(out)
ℓ,𝑖 | − 𝐼,(8)
^𝛼𝑃(𝑂) = 3|𝑃|^𝑥𝑃(𝑂),1
3|𝑃|>2˜𝜖and |^𝑥𝑃(𝑂)|>2·3|𝑃|/2√˜𝜖𝑄:𝑎𝑄̸=0 |𝑎𝑄|,
0,otherwise.(9)
The computation of ^𝑥𝑃(𝑂)and ^𝛼𝑃(𝑂)can both be done classically. The basic idea of ^𝛼𝑃(𝑂)is to set the
coefficient 3|𝑃|^𝑥𝑃(𝑂)to zero when the influence of Pauli observable 𝑃is negligible. Given an 𝑛-qubit state
𝜌, the algorithm outputs
ℎ(𝜌, 𝑂) =
𝑃:|𝑃|≤𝑘
^𝛼𝑃(𝑂) tr(𝑃 𝜌).(10)
With a proper implementation, the computational time is 𝒪(𝑘𝑛𝑘𝑁). Note that, to make predictions, the ML
algorithm only needs the 𝑘-body reduced density matrices (𝑘-RDMs) of 𝜌. The 𝑘-RDMs of 𝜌can be efficiently
obtained by performing randomized Pauli measurement on 𝜌and using the classical shadow formalism [46,47].
Except for this step, which may require quantum computation, all other steps of the ML algorithm only
requires classical computation. Hence, if the 𝑘-RDMs of 𝜌can be computed classically, then we have a
classical ML algorithm that can predict an arbitrary quantum process ℰafter learning from data.
B. Rigorous guarantee
To measure the prediction error of the ML model, we consider the average-case prediction performance
under an arbitrary 𝑛-qubit state distribution 𝒟invariant under single-qubit Clifford gates, which means that
the probability distribution 𝑓𝒟(𝜌)of sampling a state 𝜌is equal to 𝑓𝒟(𝑈𝜌𝑈†)of sampling 𝑈𝜌𝑈†for any
single-qubit Clifford gate 𝑈. We call such a distribution locally flat.
Theorem 1 (Learning an unknown quantum process).Given 𝜖, 𝜖′= Θ(1) and a training set 𝑆𝑁(ℰ)of size
𝑁=𝒪(log 𝑛)as specified in Eq. (7). With high probability, the ML model can learn a function ℎ(𝜌, 𝑂)from
摘要：

LearningtopredictarbitraryquantumprocessesHsin-YuanHuang,1SitanChen,2andJohnPreskill1,31InstituteforQuantumInformationandMatterandDepartmentofComputingandMathematicalSciences,Caltech,Pasadena,CA,USA2DepartmentofElectricalEngineeringandComputerSciences,UCBerkeley,Berkeley,CA,USA3AWSCenterforQuantumCo...

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