Robust data encodings for quantum classifiers

QTML 2019

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Ryan LaRose, Brian Coyle

arXiv:work-in-progress

Motivation: Data representation (encoding) is critical

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“Science is representation learning by humans. Deep learning is representation learning by machines.”

-- Lex Fridman, Journal of Academic Twitter.

Motivation: Data representation (encoding) is critical

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How classical data is encoded in a quantum state is crucial for learning.

There is a tradeoff between robustness and learnability with quantum encodings of classical data.

Outline

1. Models for quantum classification
2. Data encoding & learnability
3. Noise and quantum channels
4. Robustness results
5. Conclusions

Outline

• Models for quantum classification
• Data encoding & learnability
• Noise and quantum channels
• Robustness results
• Conclusions

Classification problems in machine learning

Input: Feature vectors and labels

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Output: “Intelligent” machine that correctly classifies all feature vectors

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and can make new (correct) predictions on data it was not trained on.

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Quantum classifiers

Input: Feature vectors and labels

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Output: “Intelligent” quantum machine that correctly classifies all feature vectors

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and can make new (correct) predictions on data it was not trained on.

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Common models for quantum classification

Common models for quantum classification

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1. Encode data point in a quantum state
2. Evolve with a trainable ansatz
3. Measure a single qubit to get a label {0, 1}

Common models for quantum classification

Common models for quantum classification

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• Encode data point in a quantum state
• Evolve with a trainable ansatz
• Measure a single qubit to get a label {0, 1}

Common models for quantum classification

​

• Encode data point in a quantum state
• Evolve with a trainable ansatz
• Measure a single qubit to get a label {0, 1}

Common models for quantum classification

Common models for quantum classification

​

• Encode data point in a quantum state
• Evolve with a trainable ansatz
• Measure a single qubit to get a label {0, 1}

Common models for quantum classification

Common models for quantum classification

​

• Encode data point in a quantum state
• Evolve with a trainable ansatz
• Measure a single qubit to get a label {0, 1}

Common models for quantum classification

Common models for quantum classification

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• Encode data point in a quantum state
• Evolve with a trainable ansatz
• Measure a single qubit to get a label {0, 1}

Common models for quantum classification

Theme:

• Encode data point in a quantum state

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• Evolve with a trainable ansatz

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• Measure a single qubit to get a label {0, 1}

Common models for quantum classification

Theme:

• Encode data point in a quantum state

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• Evolve with a trainable ansatz

​

• Measure a single qubit to get a label {0, 1}

Common models for quantum classification

Theme:

• Encode data point in a quantum state

​

• Evolve with a trainable ansatz

​

• Measure a single qubit to get a label {0, 1}

Common models for quantum classification

Theme:

• Encode data point in a quantum state

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• Evolve with a trainable ansatz

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• Measure a single qubit to get a label {0, 1}

Model for a (binary) quantum classifier

We consider the model commonly discussed in literature:

Model for a (binary) quantum classifier

We consider the model commonly discussed in literature:

Outline

• Models for quantum classification
• Data encoding & learnability
• Noise and quantum channels
• Robustness results
• Conclusions

Encoding data in a quantum state

(1) Complete wavefunction encoding

Takes exp(n) time

(2) QRAM

Takes infinite time (not possible)

(3) “Quantum data”

Denies that encoding problem exists

Data encodings

Basis encoding for binary data

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where each

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Data encodings

Amplitude (wavefunction) encoding for arbitrary data

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where each

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Data encodings

We can consider parameterizations of features.

Schuld and Killoran (Phys. Rev. Lett. 122, 040504) define the tensor product angle encoding

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which encodes one feature per qubit.

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Data encodings

We can also define a dense angle encoding

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which encodes two features per qubit.

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Data encodings

Such parameterizations can be generalized to any L2 functions:

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Data encodings

Such parameterizations can be generalized to any L2 functions:

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These functions directly determine the learnable decision boundaries of the model. In particular, the decision boundary can be found from

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Data encodings

Such parameterizations can be generalized to any L2 functions:

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These functions directly determine the learnable decision boundaries of the model. In particular, the decision boundary can be found from

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For a single qubit classifier, this becomes

Data encodings

Such parameterizations can be generalized to any L2 functions:

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These functions directly determine the learnable decision boundaries of the model. In particular, the decision boundary can be found from

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For a single qubit classifier, this becomes

Learnability of encodings

Single qubit encoding of two features

Dense angle encoding

Wavefunction encoding

Learnability of encodings

Single qubit encoding of two features

Data encodings

From a hardness/advantage perspective, it's a good idea to encode data with circuits that are hard to simulate classically.

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Supervised learning with quantum enhanced feature spaces, Nature. vol. 567, pp. 209-212 (2019)

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Key properties of data encodings

1. For a given encoding, what decision boundaries are possible to learn?

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• For a given encoding, how robust is the classifier to noise?

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Robustness definition

Let denote the predicted label by the quantum classifier for data point .

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Let denote a noise channel.

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We say that the classifier is robust to the noise channel if and only if

Outline

• Models for quantum classification
• Data encoding & learnability
• Noise and quantum channels
• Robustness results
• Conclusions

Noise in quantum systems

Noise occurs due to interactions between a principal quantum system and its environment.

Physically,

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Noise in quantum systems

Noise occurs due to interactions between a principal quantum system and its environment.

Physically,

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We often use the equivalent, more convenient operator-sum representation

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where

Common models for noise

Depolarizing noise:

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Dephasing noise:

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Common models for noise

Pauli noise:

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Common models for noise

Pauli noise:

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Amplitude damping:

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Common models for noise

Pauli noise:

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Amplitude damping:

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Measurement noise:

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where is the probability of getting outcome k given input l.

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Outline

• Models for quantum classification
• Data encoding & learnability
• Noise and quantum channels
• Robustness results
• Conclusions

Two regimes

1. Ideal data encoding

This characterizes (mostly) properties of the model.

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• Noisy data encoding

This characterizes (mostly) properties of the data encoding

Robustness to Pauli channels

Result 1: The classifier is robust to Pauli noise

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if

Proof:

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Robustness to Pauli channels

Result 1: The classifier is robust to Pauli noise

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if

Proof:

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Robustness to Pauli channels

Result 1: The classifier is robust to Pauli noise

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if

Proof:

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Robustness to Pauli channels

Result 1: The classifier is robust to Pauli noise

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if

Proof:

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If , then

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Robustness to Pauli channels

Result 1: The classifier is robust to Pauli noise

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if

Proof:

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If , then

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If , then

Robustness to Pauli channels

For the wavefunction encoding

Robustness to Pauli channels

For the dense angle encoding

Robustness to Pauli channels

Corollary 1: Suppose the classification scheme is modified to measure in the X basis, i.e.

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Then, the classifier is robust if

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Robustness to Pauli channels

Corollary 1: Suppose the classification scheme is modified to measure in the X basis, i.e.

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Then, the classifier is robust if

Corollary 2: Suppose the classification scheme is modified to measure in the Y basis, i.e.

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Then, the classifier is robust if

Unconditional robustness to dephasing

Result 2: The classifier is unconditionally robust to dephasing noise.

Proof:

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Unconditional robustness to dephasing

Result 2: The classifier is unconditionally robust to dephasing noise.

Proof:

1. Corollary of Pauli channel robustness.
2. Direct proof

Unconditional robustness to depolarizing noise

Result 3: The classifier is unconditionally robust to global depolarizing noise (at any point in the circuit).

General statement:

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Unconditional robustness to depolarizing noise

Result 3: The classifier is unconditionally robust to global depolarizing noise (at any point in the circuit).

Intuition for single qubit case:

If If

then then

Amplitude damping channel

Result 4: A data encoding

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is robust to amplitude damping noise iff

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for all feature vectors.

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Amplitude damping channel

Result 4: A data encoding

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is robust to amplitude damping noise iff

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for all feature vectors.

Can this be achieved? i.e., do there exist such functions f and g?

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Amplitude damping channel

Result 4: A data encoding

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is robust to amplitude damping noise iff

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for all feature vectors.

Can this be achieved? i.e., do there exist such functions f and g?

Yes.

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Amplitude damping channel

Proof of Result 4: For the noisy classifier,

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Amplitude damping channel

Proof of Result 4: For the noisy classifier,

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Suppose

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Amplitude damping channel

Proof of Result 4: For the noisy classifier,

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Suppose

Because and , we certainly have

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Amplitude damping channel

Proof of Result 4: For the noisy classifier,

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Suppose

Because and , we certainly have

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That is, noisy classification of features labelled 0 is robust.

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Amplitude damping channel

Proof of Result 4: For the noisy classifier,

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Amplitude damping channel

Proof of Result 4: For the noisy classifier,

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Suppose

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Amplitude damping channel

Proof of Result 4: For the noisy classifier,

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Suppose

We require

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Amplitude damping channel

Proof of Result 4: For the noisy classifier,

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Suppose

We require

Using resolution of the identity we arrive at

Amplitude damping channel

Proof of Result 4: For the noisy classifier,

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Suppose

We require

Using resolution of the identity we arrive at

Amplitude damping channel

Wavefunction encoding with amplitude damping noise

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Amplitude damping channel

Wavefunction encoding with amplitude damping noise

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Amplitude damping channel

Wavefunction encoding with amplitude damping noise

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Amplitude damping channel

Wavefunction encoding with amplitude damping noise

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Amplitude damping channel

Wavefunction encoding with amplitude damping noise

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Amplitude damping channel

Wavefunction encoding with amplitude damping noise

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Amplitude damping channel

Dense angle encoding with amplitude damping noise

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Amplitude damping channel

Dense angle encoding with amplitude damping noise

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Amplitude damping channel

Dense angle encoding with amplitude damping noise

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Amplitude damping channel

Dense angle encoding with amplitude damping noise

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Amplitude damping channel

Dense angle encoding with amplitude damping noise

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Amplitude damping channel

Dense angle encoding with amplitude damping noise

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Amplitude damping channel

What is a robust encoding?

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Can we always achieve robustness?

Yes.

Theorem: For any noisy quantum classifier with trace-preserving quantum operation, there exists an encoding such that the noisy classifications are robust.

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Can we always achieve robustness?

Yes.

Theorem: For any noisy quantum classifier with trace-preserving quantum operation, there exists an encoding such that the noisy classifications are robust.

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Proof: Schauder’s theorem => trace-preserving quantum channels have at least one fixed point. Fixed points => existence of a robust encoding.

Schauder’s theorem (informal): Any continuous map on a convex, compact subspace of a Hilbert space has a fixed point.

Can we always achieve robustness?

Yes.

But at the cost of learnability.

Can we always achieve robustness?

Yes.

Examples:

Bit-flip channel has fixed points [insert fixed points]

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Phase flip channel has fixed points [insert fixed points]

Outline

• Models for quantum classification
• Data encoding & learnability
• Noise and quantum channels
• Robustness results
• Conclusions

Conclusions

1. Encoding classical data in a quantum state is an important, under-studied problem.

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Conclusions

• Encoding classical data in a quantum state is an important, under-studied problem.
• Many speedups in QML algorithms rely on efficient data encoding, which may not be plausible.

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Conclusions

• Encoding classical data in a quantum state is an important, under-studied problem.
• Many speedups in QML algorithms rely on efficient data encoding, which may not be plausible.
• For “practical QML,” data encoding directly determines learnable decision boundaries.

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Conclusions

• Encoding classical data in a quantum state is an important, under-studied problem.
• Many speedups in QML algorithms rely on efficient data encoding, which may not be plausible.
• For “practical QML,” data encoding directly determines learnable decision boundaries.
• The classifier studied here exhibits robustness to a variety of common errors.

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Conclusions

• Encoding classical data in a quantum state is an important, under-studied problem.
• Many speedups in QML algorithms rely on efficient data encoding, which may not be plausible.
• For “practical QML,” data encoding directly determines learnable decision boundaries.
• The classifier studied here exhibits robustness to a variety of common errors.
• For certain error models, data can be encoded in such a way to ensure robustness while (perhaps) maintaining learnability.

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Conclusions

• Encoding classical data in a quantum state is an important, under-studied problem.
• Many speedups in QML algorithms rely on efficient data encoding, which may not be plausible.
• For “practical QML,” data encoding directly determines learnable decision boundaries.
• The classifier studied here exhibits robustness to a variety of common errors.
• For certain error models, data can be encoded in such a way to ensure robustness while (perhaps) maintaining learnability.
• A robust data encoding always exists, though (likely) at the expense of learnability.

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Conclusions

Continued work:

• Generalizing some robustness results to arbitrary qubits,
• numerical results for noisy data encoding,
• general robustness results for classes of quantum channels (unital channels, etc.)

Future directions:

• Different models of classification (distance-based classifiers) and robustness results,
• universal learning theorems,
• making the tradeoff between learnability and robustness quantitative

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Thank you for your attention.

NOTES SLIDE

• RELATIVE ENTROPIES OF QUANTUM STATES
• NIELSEN AND CHUANG EXERCISE ON FIXED POINTS OF QUANTUM CHANNELS
• THIS PAPER https://arxiv.org/pdf/1905.00857.pdf
• THIS PAPER https://pdfs.semanticscholar.org/6e9e/e38a9db7b3771a5755508b730168bdd8de69.pdf
• MONOTONICITY OF RELATIVE ENTROPY FOR UNITAL CHANNELS