Learn from noisy experiments
All actions are unknown a priori. Our goal is to learn the true physical description of each action to arbitrary accuracy.”
“practical applications:
(1) An efficient algorithm for learning Quantum channels;
(2) corollary: no protocol can learn gate-dependent Pauli noise in Clifford+T gates even under perfect state preparation and measurement.”
“Task: learning physical operations in a quantum machine.”
“Can instruct a quantum system to perform many physical operations.
- "initial states to prepare, CPTP maps to perform, and POVM measurements."
“Cannot see inside a quantum system”
/- "theorem: if operations in quantum system can be composed to explore the full Hilbert space,
then Writer can learn everything from scratch.
-/
theorem : ... := sorry
The proof utilizes Randomized measurements, random walk on Lie groups, Wigner’s theorem, and more”
- Reminds me of something in thermalization, visiting all states in the Hilbert space;
/-- "If Writer can explore the full quantum state space by composing the operations,
then every operation can be learned." -/
theorem : ... := sorry
An efficient algorithm for learning all operations up to a single unlearnable parameter 𝑓 = ⟨0𝑛|𝜌0|0𝑛⟩.”
The parameter 𝑓 is the fidelity of the noisy zero state 𝜌0, which is assumed to be close to one.
theorem f_unlearnable :
unlearnable f :=
begin
/-
Assume that the noise channel 𝒩 is the identity channel
One cannot distinguish whether the initial state or the measurement is subjected to a depolarizing channel.
The learning is not easier for non-identity channel.
So 𝑓 is still unlearnable."
-/
sorry,
end
“When one cannot explore the full state space. But all operations are approximately known. And noise in Clifford gates is gate-independent.
“For learning a noise channel on Clifford gates to fixed accuracy.
The true description of the noise can be unlearnable; for example, no benchmarking protocol can learn gate-dependent Pauli noise on Clifford+T gates. Even under perfect state preparation and measurement.
Despite not being able to learn the noise. A noisy quantum computer. That performs entangled measurements. On multiple copies of an unknown state. Can yield a large advantage. In learning properties of the state.
Compared to a noiseless device. That measures individual copies. And then processes the measurement data. Using a classical computer.”
“Error using a finite number of experiments, up to one global unitary or anti-unitary transformation. A similar criterion is often invoked in discussions of Quantum tomography;
def learnability := sorry
structure world_model : Type :=
({𝜌𝑥}𝑥∈𝒳 ) -- quantum state
({E𝑦}𝑦∈𝒴) -- quantum channel
({M𝑧}𝑧∈𝒵) -- POVM measurement
def universal (𝒲 : world_model) :=
∃ 𝑥 ∈ 𝒳, pure 𝜌𝑥 ∧
∃ 𝑦1,...,𝑦𝑘 ∈ 𝒴, E𝑦1,...E𝑦𝑘 constitute a universal set of unitary transformations ∧
∃ 𝑧 ∈ 𝒵, POVM M𝑧 has at least one element not proportional to the identity. ∧
/-- Theorem 2: Learning intrinsic descriptions -/
theorem [model class 𝒬] [each candidate world model in 𝒬 is universal] :
If the true world model is 𝒲true ∈ 𝒬, then Writer can learn the description of every action in 𝒲true to arbitrarily small error := sorry
/-- Theorem 3: Worst case complexity for predicting extrinsic behavior -/
theorem : To predict the probability of each experimental outcome to error 𝜖 for every experiment with 𝐿 maps, Writer has to perform at least Ω(|𝒳||𝒴|𝐿|𝒵|/𝜖2) experiments in the worst case,
and Writer can always achieve the task by running 𝒪̃︀(|𝒳||𝒴|𝐿|𝒵|/𝜖2) experiments := sorry
/-- Theorem 4: Predicting extrinsic behavior -/
theorem : Suppose we have found a set of unknown linearly indepen- dent states composed from {𝜌𝑥,E𝑦} and a set of unknown POVM elements composed from {E𝑦,M𝑧}, such that both sets span all the states that can be prepared in world model 𝒲.
Then, we can predict the probability of each experimental outcome to error 𝜖 for any experiment with 𝐿 maps after running 𝒪̃︀((|𝒳 | + 𝐿2|𝒴| + |𝒵|)/𝜖2) experiments."
/-- Theorem 5: Learning gate-independent noise on Clifford gates -/ theorem :
Assume bounded gate-independent noise on Clifford gates.
And bounded noise on initial zero state preparation and computational basis measurement.
All states, processes, and measurements {𝜌𝑥,E𝑦,M𝑧} can be learned
up to a single unlearnable parameter 𝑓 = ⟨0𝑛|𝜌0|0𝑛⟩ := sorry
/-- Theorem 6: Gate-dependent Pauli noise is unlearnable with Clifford+T gates -/
theorem: Consider a qubit. Suppose one can prepare |0⟩ perfectly, measure in the computational basis perfectly, and apply ..."
“Common assumptions in Randomized benchmarking:
Noise processes afflicting Clifford gates are independent of the specific gate.
This assumption applies to all elements in the Clifford group. And is used to ensure that Randomized benchmarking can extract an accurate estimate for the average gate fidelity.
More precisely, when the classical agent performs a Clifford gate 𝐶, the CPTP map implemented is E𝐶 (𝜌) = 𝒩(𝐶𝜌𝐶†), where 𝒩 is a CPTP map close to the identity that does not depend on the gate 𝐶.
Also commonly assumed: There is a noisy zero state 𝜌0 that is close to the all zero state |0𝑛⟩. And there is a noisy computational basis measurement M0 = {𝑀𝑏}𝑏∈{0,1}𝑛, where 𝑀𝑏 is close to |𝑏⟩⟨𝑏|.
We will refer to these assumptions as bounded gate-independent noise on Clifford gates, and bounded noise on initial zero state preparation and computational basis measurement.
Typically, these assumptions are expected to hold (approximately) only for a subsystem consisting of a constant number of qubits in a many-qubit quantum computer.”