Phase transition in magic with random quantum circuits�Pradeep Niroula, Christopher David White, Qingfeng Wang, Sonika Johri, Daiwei Zhu, Christopher Monroe, Crystal Noel, Michael J. Gullans

https://arxiv.org/abs/2304.10481

Presented by Daniel Strano

Overview

Overview

• This report is about an experiment to measure the effect of (Clifford) error correction on “magic” as noise in a Clifford circuit.

Overview

• This report is about an experiment to measure the effect of (Clifford) error correction on “magic” as noise in a Clifford circuit.
• The authors define at least two different measures of “magic,” (i.e. deviation from pure Clifford or stabilizer states).

Overview

• This report is about an experiment to measure the effect of (Clifford) error correction on “magic” as noise in a Clifford circuit.
• The authors define at least two different measures of “magic,” (i.e. deviation from pure Clifford or stabilizer states).
• Below a critical error rate or code rate (i.e. when error correction works) magic is suppressed or eliminated.

Overview

• This report is about an experiment to measure the effect of (Clifford) error correction on “magic” as noise in a Clifford circuit.
• The authors define at least two different measures of “magic,” (i.e. deviation from pure Clifford or stabilizer states).
• Below a critical error rate or code rate (i.e. when error correction works) magic is suppressed or eliminated.
• Above the same critical rate (i.e. when the system is too noisy for the error correction code) magic is concentrated.

Overview

• This report is about an experiment to measure the effect of (Clifford) error correction on “magic” as noise in a Clifford circuit.
• The authors define at least two different measures of “magic,” (i.e. deviation from pure Clifford or stabilizer states).
• Below a critical error rate or code rate (i.e. when error correction works) magic is suppressed or eliminated.
• Above the same critical rate (i.e. when the system is too noisy for the error correction code) magic is concentrated.
• A phase transition is observed between these two regimes.

What is “magic”?

​

​

What is “magic”? (...Or “quantum theory,” for that matter?)

(...No, not that kind.)🤷‍♀️

What is “magic”?

​

​

What is “magic”?

• “Clifford” or “stabilizer” states can be simulated efficiently on a classical computer.

​

What is “magic”?

• “Clifford” or “stabilizer” states can be simulated efficiently on a classical computer.
• These states can have superposition and entanglement, but they are not universal over all possible quantum states in the Hilbert space.

​

What is “magic”?

• “Clifford” or “stabilizer” states can be simulated efficiently on a classical computer.
• These states can have superposition and entanglement, but they are not universal over all possible quantum states in the Hilbert space.
• “Magic” is a measure of how much a general quantum state varies from being purely Clifford.

​

What is “magic”?

• “Clifford” or “stabilizer” states can be simulated efficiently on a classical computer.
• These states can have superposition and entanglement, but they are not universal over all possible quantum states in the Hilbert space.
• “Magic” is a measure of how much a general quantum state varies from being purely Clifford.
• Quantity of “magic” therefore seems to serve proxy for how theoretically difficult a state is to simulate.

​

What is “magic”?

• “Clifford” or “stabilizer” states can be simulated efficiently on a classical computer.
• These states can have superposition and entanglement, but they are not universal over all possible quantum states in the Hilbert space.
• “Magic” is a measure of how much a general quantum state varies from being purely Clifford.
• Quantity of “magic” therefore seems to serve proxy for how theoretically difficult a state is to simulate.
• If there’s no “magic,” there’s no quantum advantage.

How do we quantify “magic”?

​

​

​

How do we quantify “magic”?

• Multiple measures of “magic” have been used, (like count of “T” gates in Clifford+T circuits,) but some are less rigorous as quantitative metrics.

​

​

How do we quantify “magic”?

• Multiple measures of “magic” have been used, (like count of “T” gates in Clifford+T circuits,) but some are less rigorous as quantitative metrics.
• This reports suggests using the “second stabilizer Rényi entropy” (SSRE).

​

How do we quantify “magic”?

• Multiple measures of “magic” have been used, (like count of “T” gates in Clifford+T circuits,) but some are less rigorous as quantitative metrics.
• This reports suggests using the “second stabilizer Rényi entropy” (SSRE).
• Measures how spread out the state’s density matrix is when expanded in the basis of Pauli operators

​

How do we quantify “magic”?

• Multiple measures of “magic” have been used, (like count of “T” gates in Clifford+T circuits,) but some are less rigorous as quantitative metrics.
• This reports suggests using the “second stabilizer Rényi entropy” (SSRE).
• Measures how spread out the state’s density matrix is when expanded in the basis of Pauli operators
• For an exact Clifford state, Pauli operators “stabilize” the state, so SSRE=0

​

How do we quantify “magic”?

• Multiple measures of “magic” have been used, (like count of “T” gates in Clifford+T circuits,) but some are less rigorous as quantitative metrics.
• This reports suggests using the “second stabilizer Rényi entropy” (SSRE).
• Measures how spread out the state’s density matrix is when expanded in the basis of Pauli operators
• For an exact Clifford state, Pauli operators “stabilize” the state, so SSRE=0
• “A Haar state on N qubits… has approximately equal weight on all Pauli operators, so it is nearly maximally spread out and the SSRE, defined as M₂(ρ) = − log (1/2^N) ∑_P∈PTr(ρP )⁴ for N qubits, is proportional to N.”

How do we quantify “magic”? (continued)

• Alternatively, the report also considers “basis-minimized measurement entropy,” as a measure of “magic.”

​

How do we quantify “magic”? (continued)

• Alternatively, the report also considers “basis-minimized measurement entropy,” as a measure of “magic.”
• “...defined as the entropy of the Born probability distribution of measurement outcomes, minimized over the finite set of possible stabilizer measurement bases.”

​

How do we quantify “magic”? (continued)

• Alternatively, the report also considers “basis-minimized measurement entropy,” as a measure of “magic.”
• “...defined as the entropy of the Born probability distribution of measurement outcomes, minimized over the finite set of possible stabilizer measurement bases.”
• In other words, we check all possible Pauli bases for the minimum Born probability distribution entropy of measurement outcomes.

​

How do we quantify “magic”? (continued)

• Alternatively, the report also considers “basis-minimized measurement entropy,” as a measure of “magic.”
• “...defined as the entropy of the Born probability distribution of measurement outcomes, minimized over the finite set of possible stabilizer measurement bases.”
• In other words, we check all possible Pauli bases for the minimum Born probability distribution entropy of measurement outcomes.
• For a stabilizer state, there is always some Pauli basis with 0 entropy.

​

How do we quantify “magic”? (continued)

• Alternatively, the report also considers “basis-minimized measurement entropy,” as a measure of “magic.”
• “...defined as the entropy of the Born probability distribution of measurement outcomes, minimized over the finite set of possible stabilizer measurement bases.”
• In other words, we check all possible Pauli bases for the minimum Born probability distribution entropy of measurement outcomes.
• For a stabilizer state, there is always some Pauli basis with 0 entropy.
• Non-increasing for stabilizer operations; subadditive for product states

Experimental design

Experimental design

Experimental design

• Start with a Clifford circuit that interleaves gates of 1 and 2 qubits.

Experimental design

• Start with a Clifford circuit that interleaves gates of 1 and 2 qubits.
• Apply a single-qubit Z-basis rotation on each qubit, with α between 0 and π/2.

Experimental design

• Start with a Clifford circuit that interleaves gates of 1 and 2 qubits.
• Apply a single-qubit Z-basis rotation on each qubit, with α between 0 and π/2.
• Run the inverse of the original Clifford circuit.

Experimental design

• Start with a Clifford circuit that interleaves gates of 1 and 2 qubits.
• Apply a single-qubit Z-basis rotation on each qubit, with α between 0 and π/2.
• Run the inverse of the original Clifford circuit.
• Measure (N-K) syndrome qubits out of N qubits.

Experimental design

• Start with a Clifford circuit that interleaves gates of 1 and 2 qubits.
• Apply a single-qubit Z-basis rotation on each qubit, with α between 0 and π/2.
• Run the inverse of the original Clifford circuit.
• Measure (N-K) syndrome qubits out of N qubits.
• K qubits are logical; (N-K) are error correction syndrome.

Experimental design

• Start with a Clifford circuit that interleaves gates of 1 and 2 qubits.
• Apply a single-qubit Z-basis rotation on each qubit, with α between 0 and π/2.
• Run the inverse of the original Clifford circuit.
• Measure (N-K) syndrome qubits out of N qubits.
• K qubits are logical; (N-K) are error correction syndrome.
• Apply error correction syndrome, which either concentrates toward a grid of Clifford states, with 0 “magic”/entropy, or pushes off that grid.

Experimental design

• Start with a Clifford circuit that interleaves gates of 1 and 2 qubits.
• Apply a single-qubit Z-basis rotation on each qubit, with α between 0 and π/2.
• Run the inverse of the original Clifford circuit.
• Measure (N-K) syndrome qubits out of N qubits.
• K qubits are logical; (N-K) are error correction syndrome.
• Apply error correction syndrome, which either concentrates toward a grid of Clifford states, with 0 “magic”/entropy, or pushes off that grid.
• Based on some assumptions about tomography calculated via the syndrome, we can quantify “magic.”

Experimental design

• Start with a Clifford circuit that interleaves gates of 1 and 2 qubits.
• Apply a single-qubit Z-basis rotation on each qubit, with α between 0 and π/2.
• Run the inverse of the original Clifford circuit.
• Measure (N-K) syndrome qubits out of N qubits.
• K qubits are logical; (N-K) are error correction syndrome.
• Apply error correction syndrome, which either concentrates toward a grid of Clifford states, with 0 “magic”/entropy, or pushes off that grid.
• Based on some assumptions about tomography calculated via the syndrome, we can quantify “magic.”
• Experiment was run on 16 qubits of IonQ’s Aria processor.

How does the experiment measure magic?

​

​

​

​

How does the experiment measure magic?

• Change basis and measure the entire register.

​

​

​

How does the experiment measure magic?

• Change basis and measure the entire register.
• We effectively need to perform single-qubit tomography, from the results.

​

​

How does the experiment measure magic?

• Change basis and measure the entire register.
• We effectively need to perform single-qubit tomography, from the results.
• Post-selection on syndrome outcomes is too expensive, but set of possible actions on logical qubit is much smaller than the set of possible syndromes.

​

How does the experiment measure magic?

• Change basis and measure the entire register.
• We effectively need to perform single-qubit tomography, from the results.
• Post-selection on syndrome outcomes is too expensive, but set of possible actions on logical qubit is much smaller than the set of possible syndromes.
• Consider different syndromes equivalent, when they have the same effect.

​

How does the experiment measure magic?

• Change basis and measure the entire register.
• We effectively need to perform single-qubit tomography, from the results.
• Post-selection on syndrome outcomes is too expensive, but set of possible actions on logical qubit is much smaller than the set of possible syndromes.
• Consider different syndromes equivalent, when they have the same effect.
• To mitigate incoherent errors, project the single-qubit density matrix and rotate so that it has maximum eigenvalue along the basis. (Unrelated work, but this essentially Qrack’s “Schmidt decomposition rounding parameter”!)

How does the experiment measure magic?

• Change basis and measure the entire register.
• We effectively need to perform single-qubit tomography, from the results.
• Post-selection on syndrome outcomes is too expensive, but set of possible actions on logical qubit is much smaller than the set of possible syndromes.
• Consider different syndromes equivalent, when they have the same effect.
• To mitigate incoherent errors, project the single-qubit density matrix and rotate so that it has maximum eigenvalue along the basis. (Unrelated work, but this essentially Qrack’s “Schmidt decomposition rounding parameter”!)
• This rotation is done with tomography on the syndrome class, and average magic is calculated per syndrome class.

Results - vanishing rate code

​

​

​

​

​

Results - vanishing rate code

• α varies from 0 to π/2, Clifford at the extremes, with peaked magic at 1/√N

​

​

​

​

Results - vanishing rate code

• α varies from 0 to π/2, Clifford at the extremes, with peaked magic at 1/√N
• For a single logical qubit, we make a simplifying assumption that exactly two errors give rise to a syndrome, n_a and n_b, drawn from a binomial distribution.

​

​

​

Results - vanishing rate code

• α varies from 0 to π/2, Clifford at the extremes, with peaked magic at 1/√N
• For a single logical qubit, we make a simplifying assumption that exactly two errors give rise to a syndrome, n_a and n_b, drawn from a binomial distribution.
• Ratio of amplitudes is [tan(π/2-α)]^(n_a-n_b)≈(π/2-α)(n_a-n_b)

​

​

Results - vanishing rate code

• α varies from 0 to π/2, Clifford at the extremes, with peaked magic at 1/√N
• For a single logical qubit, we make a simplifying assumption that exactly two errors give rise to a syndrome, n_a and n_b, drawn from a binomial distribution.
• Ratio of amplitudes is [tan(π/2-α)]^(n_a-n_b)≈(π/2-α)(n_a-n_b)
• SSRE is therefore M₂≈(π/2-α)²(n_a-n_b)²

​

Results - vanishing rate code

• α varies from 0 to π/2, Clifford at the extremes, with peaked magic at 1/√N
• For a single logical qubit, we make a simplifying assumption that exactly two errors give rise to a syndrome, n_a and n_b, drawn from a binomial distribution.
• Ratio of amplitudes is [tan(π/2-α)]^(n_a-n_b)≈(π/2-α)(n_a-n_b)
• SSRE is therefore M₂≈(π/2-α)²(n_a-n_b)²
• Since this is binomial, M₂ ∝ N (π/2 − α)2 = f ((π/2 − α)√N )

​

Results - vanishing rate code

• α varies from 0 to π/2, Clifford at the extremes, with peaked magic at 1/√N
• For a single logical qubit, we make a simplifying assumption that exactly two errors give rise to a syndrome, n_a and n_b, drawn from a binomial distribution.
• Ratio of amplitudes is [tan(π/2-α)]^(n_a-n_b)≈(π/2-α)(n_a-n_b)
• SSRE is therefore M₂≈(π/2-α)²(n_a-n_b)²
• Since this is binomial, M₂ ∝ N (π/2 − α)2 = f ((π/2 − α)√N )
• On average:〈M₂〉= f ((π/2 − α)√N )

Results - vanishing rate code

Results - constant rate codes

​

​

​

​

Results - constant rate codes

• (Finite rate definition: K, logical qubits, scales as K = rN for N physical qubits)

​

​

​

Results - constant rate codes

• (Finite rate definition: K, logical qubits, scales as K = rN for N physical qubits)
• “...the finite-magic critical region displayed by the vanishing-rate code becomes an extended magical phase.”

​

​

Results - constant rate codes

• (Finite rate definition: K, logical qubits, scales as K = rN for N physical qubits)
• “...the finite-magic critical region displayed by the vanishing-rate code becomes an extended magical phase.”
• “The scaling collapse indicates that the transition from non-magical to magical is indeed a phase transition, not a crossover.”

​

Results - constant rate codes

• (Finite rate definition: K, logical qubits, scales as K = rN for N physical qubits)
• “...the finite-magic critical region displayed by the vanishing-rate code becomes an extended magical phase.”
• “The scaling collapse indicates that the transition from non-magical to magical is indeed a phase transition, not a crossover.”
• SSRE is expensive to calculate, so the authors use the conditional entropy of logical state as a diagnostic to study the phase transition.

Results - constant rate codes

Discussion

​

​

Discussion

• Below a critical error rate or code rate, error correction suppresses magic.

​

Discussion

• Below a critical error rate or code rate, error correction suppresses magic.
• Above the threshold, error correction concentrates magic.

​

Discussion

• Below a critical error rate or code rate, error correction suppresses magic.
• Above the threshold, error correction concentrates magic.
• A phase transition is observed between, in the constant rate case.

Discussion

• Below a critical error rate or code rate, error correction suppresses magic.
• Above the threshold, error correction concentrates magic.
• A phase transition is observed between, in the constant rate case.
• The authors raise the possibility of leveraging noise for controlled “magic.”

Discussion

• Below a critical error rate or code rate, error correction suppresses magic.
• Above the threshold, error correction concentrates magic.
• A phase transition is observed between, in the constant rate case.
• The authors raise the possibility of leveraging noise for controlled “magic.”�
• Dan the presenter finds it counterintuitive: the better our error correction is working, the easier it is to efficiently classically simulate our system.

Discussion

• Below a critical error rate or code rate, error correction suppresses magic.
• Above the threshold, error correction concentrates magic.
• A phase transition is observed between, in the constant rate case.
• The authors raise the possibility of leveraging noise for controlled “magic.”�
• Dan the presenter finds it counterintuitive: the better our error correction is working, the easier it is to efficiently classically simulate our system.
• The study does not attempt to inject controlled “magic states” into the system for universal quantum computation, but rather only to act as noise.

Thank you!

Contact: dan@unitary.fund