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Characterization of addressability by simultaneous randomized benchmarking
Jay M. Gambetta, A. D. Corcoles, S. T. Merkel, B. R. Johnson, John A. Smolin, Jerry M. Chow, Colm A. Ryan, Chad Rigetti, S. Poletto, Thomas A. Ohki, Mark B. Ketchen, M. Steffen
TL;DR
Assessing model fit is complicated by uncertainty about the distribution of randomized sequence fidelities. The paper uses nonlinear least-squares fitting and finds reduced chi-square values close to or below one, supporting a good fit to the data.
Problem
Goodness-of-fit assessment is limited by open questions about the distribution of randomized sequence fidelities.
Method
The data are fit to a three-parameter model using standard nonlinear least-squares regression, with uncertainties estimated from the Jacobian.
Results
Reduced chi-square values for both samples’ fits are close to or below one, leading the authors to conclude that the model fits the data well.
Takeaways & Limitations
The reported fit statistics support using the three-parameter model to describe the data for both samples.
Abstract
from arXiv · showhide
The control and handling of errors arising from cross-talk and unwanted interactions in multi-qubit systems is an important issue in quantum information processing architectures. We introduce a benchmarking protocol that provides information about the amount of addressability present in the system and implement it on coupled superconducting qubits. The protocol consists of randomized benchmarking each qubit individually and then simultaneously, and the amount of addressability is related to the difference of the average gate fidelities of those experiments. We present the results on two similar samples with different amounts of cross-talk and unwanted interactions, which agree with predictions based on simple models for the amount of residual coupling.
Supplementary material for ‘Characterization of addressability by simultaneous · PARAMETER ESTIMATION: A STATISTICAL ANALYSIS
The supplementary analysis fits experimental data with a three-parameter nonlinear least-squares model and estimates uncertainties using a linearized Jacobian at 68% confidence. Reduced chi-square values for both samples are generally close to or below one, supporting the fitted model.
- PARAMETER ESTIMATION: A STATISTICAL ANALYSIS: Three-parameter fits use standard nonlinear least-squares regression through Matlab’s nlinfit routine.The analysis identifies uncertainties from the Jacobian at the best-fit point under a linear assumption.
- PARAMETER ESTIMATION: A STATISTICAL ANALYSIS: 68% (1σ) confidence intervals are obtained with Matlab’s nlparci routine.The confidence setting is explicitly identified as 68% (1σ).
- PARAMETER ESTIMATION: A STATISTICAL ANALYSIS: Goodness-of-fit assessment is complicated by open questions about the underlying distribution of sequence fidelities.The passage states that there is no gua…
- PARAMETER ESTIMATION: A STATISTICAL ANALYSIS: N denotes the number of experimentally measured quantities xi, while µi is the model’s expected value and σ2The supplied passage ends mid-definition.
- PARAMETER ESTIMATION: A STATISTICAL ANALYSIS: For the truncation analysis, N is the number of truncations, xi is averaged sequence fidelity, and σi is its standard error.The sequence fidelity is averaged over different random Clifford strings.
- PARAMETER ESTIMATION: A STATISTICAL ANALYSIS: 1.567, 0.809, 1.369, 1.639, and 1.189 are sample a’s reduced chi-square values for the reported fit parameters.Sample a uses 32 truncations and has 29 degrees of freedom.
- PARAMETER ESTIMATION: A STATISTICAL ANALYSIS: 0.638, 1.609, 0.156, 0.489, and 0.280 are sample b’s corresponding reduced chi-square values.Sample b uses 11 truncations and has 8 degrees of freedom.
- PARAMETER ESTIMATION: A STATISTICAL ANALYSIS: All reported reduced chi-square values are close to or less than one, leading the analysis to conclude that the model isThe supplied passage ends before completing the conclusion.
REVIEW OF GROUP THEORY
This section reviews finite-group representation theory and shows how irreducible decompositions support twirling. It introduces Schur’s lemma, its twirling corollaries, and tensor-product representations for composite systems.
- Representation theory of finite groups: A group representation maps elements to complex matrices while preserving the group operation under matrix multiplication.Representations need not be one-to-one; the trivial representation maps every element to 1.
- Representation theory of finite groups: Representations are reducible when a similarity transform makes every represented group element block diagonal; otherwise they are irreducible.Every reducible representation decomposes into irreducible components.
- Schur’s lemma: Schur’s lemma states that matrices commuting with an irreducible representation are scalar multiples of the identity, while intertwiners between irreducible representations are zero or establish equivalence.These are presented as two complementary forms of Schur’s lemma.
- Twirling corollaries: Schur’s lemma yields increasingly general twirling corollaries for irreducible, distinct direct-sum, and fully reducible representations.The resulting formulas characterize group averages using identity operators, projectors, and multiplicity-copy subspaces.
- Product-group representations: For a product group G × G, tensor products of irreducible representations of G form its irreducible representations, relevant to composite Hilbert spaces.The construction includes all pairwise tensor products σj ⊗ σk.
Decomposition of some useful unitary groups · Pauli group
For n-qubit Pauli conjugation channels, the Pauli-transfer matrices are diagonal and form a group under composition, even though the Pauli matrices themselves do not. Their multiplication is represented by binary-vector addition, yielding the regular representation of Z_2^2n and its decomposition into all irreducible representations.
- Pauli group: Pauli conjugation channels are defined by Λ_k(ρ) = P_kρP_k for n-qubit Pauli operators P_k.The convention includes P_0 = I⊗n.
- Pauli group: Because Pauli operators commute or anticommute, conjugation maps each P_j to ±P_j.This property determines the diagonal structure of the transfer matrices.
- Pauli group: Encoding {I, X, Y, Z} as bit pairs (vw) provides two binary vectors (v, w) for each Pauli operator and describes the transfer-matrix elements.The passage introduces the binary encoding used to derive the general representation.
- Pauli group: All transfer matrices are diagonal, with half their entries equal to 1 and half equal to −1, while the identity maps to all 1s.This describes the diagonal pattern of the matrices associated with Pauli channels.
- Pauli group: Although the 4^n Pauli matrices do not form a group under matrix multiplication, their transfer matrices do form a group under composition.The group structure follows because unitary-channel conjugation is insensitive to global phase.
- Pauli group: Transfer-matrix multiplication follows (v(k), w(k)) = (v(j) ⊕ v(i), w(j) ⊕ w(i)), representing Z_2^2n.The binary-vector rule gives the group multiplication law for the representation.
- Pauli group: The transfer matrices constitute the regular representation of Z_2^2n, defined on the vector space of the group elements themselves.This identifies the representation underlying the Pauli-channel transfer matrices.
- Pauli group: For Z_2^2n, the regular representation is the direct sum of all 2^2n distinct irreducible representations.This gives the representation’s decomposition into irreducible components.
Clifford group
The Clifford group normalizes the Pauli group, so its Pauli-transfer-matrix representation preserves the identity and acts irreducibly on the remaining Pauli subspace. For independently controlled subsystems, these representations combine as tensor-product decompositions of trivial and irreducible components.
- Definition and PTM structure: The Clifford group is the normalizer of the Pauli group under unitary-channel conjugation, mapping each Pauli operator to another up to sign.This normalizer action permutes Pauli elements while allowing a ± sign.
- Definition and PTM structure: Clifford-group PTMs contain one 1 or −1 in each row and column, with all remaining entries zero, subject to physicality constraints.Commutator preservation and complete positivity impose additional restrictions on which such matrices are physical.
- Representation decomposition: The Clifford-group PTM representation decomposes as I ⊕ σ, with the identity operator forming the trivial invariant subspace and σ irreducible on the remaining Paulis.The identity P0 = I is left invariant by every Clifford channel.
- Representation decomposition: For independently acting Clifford operators on n subsystems, the irreducible representations have the form (I + σ)n.For two subsystems, the components include II, Iσ, σI, and σσ, corresponding to zero, one, or two nonidentity Pauli factors.
Unitary group
The full unitary group has essentially the same structure as the Clifford group in its PTM representation: the identity is stabilized while the remaining subspace is irreducible.
- Unitary group: The PTM representation of a general unitary channel has the form I ⊕ σ, with the identity stabilized and the remaining subspace irreducible.This structural similarity follows because the PTM mapping is quadratic in unitary matrices and the Clifford group forms a 2-design of the unitary group.
Twirling
The section defines twirling a channel over a group of unitary channels and states that its representation follows in the R representation. It attributes the resulting structure to the Pauli transfer matrices and Corollaries 3, 4, and 5.
- Twirling is defined by averaging a channel Λ over a group of unitary channels G = {U1, U2, . . .}.
- The twirled channel is expressed in the R representation.
- The results of twirling follow from the representation structure of the Pauli transfer matrices and Corollaries 3, 4, and 5.
Pauli twirling
Pauli twirling removes off-diagonal PTM elements, leaving the diagonal subspace corresponding exactly to Pauli channels. This follows because the Pauli-group PTM representation decomposes into distinct one-dimensional irreducible representations.
- Pauli twirling: The Pauli-group PTM representation is a direct sum of 4n 1-dimensional distinct irreducible representations.Each Pauli operator is stabilized by every group element because the Pauli group is Abelian, making the twirling result follow from Corollary 2.
- Pauli twirling: Pauli twirling leaves only the diagonal elements of the PTM representation, exactly characterizing Pauli channels.The resulting channels have the form P.
Clifford twirling
Clifford twirling reduces the relevant representation to identity and nonidentity sectors. For trace-preserving maps, this yields depolarizing channels whose survival probability is determined by Tr(RΛ)−1.
- The Clifford group representation splits into two irreducible blocks: the identity and everything else.
- For any trace-preserving map, the representation element mapping the identity to itself is unity.
- The identity component is 1 for i = 0, while the other components are (Tr(RΛ) −1)/(d2 −1).
- The resulting channels are depolarizing channels, Λdep(ρ) = αρ + (1 −α)I, with survival probability α = Tr(RΛ)−1.
Subsytem Clifford twirling
For a two-qubit system, the section analyzes Clifford twirling over C ⊗ C, C ⊗ I, and I ⊗ C, using irreducible PTM representations and symmetry to derive the twirled form. Twirling over C ⊗ I is more complex because its PTM contains representation multiplicities, but each configuration of the untwirled qubit undergoes a single-qubit twirl.
- C ⊗ C twirling: The analysis considers two-qubit Clifford twirling over C ⊗ C, C ⊗ I, and I ⊗ C.For C ⊗ C, the PTM decomposes into four distinct irreducible representations: II, Iσ, σI, and σσ.
- C ⊗ C twirling: For C ⊗ C, projectors PII, PIσ, PσI, and Pσσ are defined for the corresponding irreducible subspaces.The supplied passage identifies these projectors but does not provide the completed twirl expression.
- C ⊗ I twirling: Twirling over C ⊗ I is more complicated because distinct irreducible representations occur with multiplicity in the PTM.The representations are labeled by the first-qubit and second-qubit components, including singleton and three-element sets such as σσ,I = {XI, YI, ZI}.
- C ⊗ I twirling: Symmetry forces matrix elements with unequal indices to vanish under the C ⊗ I twirl.A Clifford operation can preserve one Pauli operator while negating another, producing cancellation after substituting Um → UmC.
- C ⊗ I twirling: For the nonzero components, Clifford twirling preserves identity and uniformly redistributes X, Y, and Z.The resulting interpretation is that each configuration of the untwirled qubit experiences a single-qubit twirl.
Unitary twirling
Twirling over the entire unitary group produces exactly the same result as twirling over the Clifford group.
- Unitary twirling: Entire-unitary-group twirling yields exactly the same result as Clifford-group twirling.
- Unitary twirling: The unitary-group and Clifford-group twirling procedures are equivalent in their resulting output.
- Unitary twirling: Replacing Clifford-group twirling with twirling over the entire unitary group does not change the result.