Estimating the Coherence of Noise

Noise mechanisms in quantum systems can be broadly characterized as either coherent (i.e., unitary) or incoherent. For a given fixed average error rate, coherent noise mechanisms will generally lead to a larger worst-case error than incoherent noise. We show that the coherence of a noise source can be quantified by the unitarity, which we relate to the average change in purity averaged over input pure states. We then show that the unitarity can be efficiently estimated using a protocol based on randomized benchmarking that is efficient and robust to state-preparation and measurement errors. We also show that the unitarity provides a lower bound on the optimal achievable gate infidelity under a given noisy process.

Noise mechanisms in quantum systems can be broadly characterized as either coherent (i.e., unitary) or incoherent. For a given fixed average error rate, coherent noise mechanisms will generally lead to a larger worst-case error than incoherent noise. We show that the coherence of a noise source can be quantified by the unitarity, which we relate to the average change in purity averaged over input pure states. We then show that the unitarity can be efficiently estimated using a protocol based on randomized benchmarking that is efficient and robust to state-preparation and measurement errors.
To harness the advantages of quantum information processing, quantum systems have to be controlled to within some maximum threshold error. Certifying whether the error is below the threshold is possible by performing full quantum process tomography [1,2] or the efficient method of direct fidelity estimation [3,4], however, both these approaches are sensitive to state-preparation and measurement errors (SPAM) [5].
Randomized benchmarking [6][7][8][9][10][11] has been developed as an efficient method for estimating the average infidelity of noise to the identity. However, the worst-case error, as quantified by the diamond distance from the identity, can be more relevant to determining whether an experimental implementation is at the threshold for fault-tolerant quantum computation [12]. The best possible bound on the worst-case error (without further assumptions on the noise) scales as the square root of the infidelity and can be orders of magnitude greater than the reported average error [13,14]. However, if the noise is known to be diagonal in the Pauli-Liouville representation (and is therefore incoherent if the error is small enough), the worst-case error is directly proportional to the average error [11] and hence vastly improves on the general bound.
Consequently, quantifying the coherence of the noise may help to significantly reduce the uncertainty in the worst-case error compared to knowing only the average-case error, and hence to certify that the experimental implementation is close to the threshold for fault-tolerance.
Furthermore, the spread of survival probabilities over random sequences of a fixed length in randomized benchmarking experiments is dominated by unitary noise [7,13]. Since a larger spread of survival probabilities requires sampling a larger number of sequences, having an estimate of the coherence of the noise may also enable better choices of experimental designs to obtain more reliable estimates of the average gate infidelity.
Randomized benchmarking is also emerging as a useful tool for diagnosing the noise in an experiment [15,16], which can then be used to optimize the implementation of gates by varying the experimental design. In this spirit, an experimental protocol for characterizing the coherence of a noise channel will be an important tool as the quest to build a faulttolerant quantum computer progresses.
In this paper, we present a protocol for estimating a particular quantification of the coherence of noise, which we term the unitarity, in the experimental implementation of a unitary 2design. Our protocol is efficient and robust against SPAM, and is a minor modification of randomized benchmarking. The unitarity is defined as the average change in the purity of a pure state after applying the noise channel, with the contributions due to the identity component subtracted off; see Eq. (4). We show that the unitarity is invariant under unitary gates and attains its maximal value if and only if the noise is unitary. Furthermore, we show that the unitarity can be combined with the average gate fidelity to quantify how far a noise channel is from depolarizing noise.
Our approach to quantifying coherence complements other recent work on quantifying coherence since we focus on the coherence of quantum operations rather than the coherence of quantum states relative to a preferred basis [17].

I. DEFINING UNITARITY
We begin by defining the unitarity of a noise channel E : , that is, a completely positive (CP) linear map that takes quantum states to quantum states. The purity of a quantum state ρ is Trρ † ρ ∈ [0, 1] with Trρ † ρ = 1 if and only if ρ is a pure state. An initial candidate for a definition of the unitarity of E is that is, as the purity of the output states averaged over all pure state inputs. However, this definition is problematic, since it would lead to the nonunital state-preparation channel having the same value of unitarity as a unitary channel, even though it does not preserve coherent superpositions. Similarly, the (trace-decreasing) filtering channel does not preserve coherent superpositions and so should have the same unitarity value as a complete depolarizing channel. Both of these problematic channels arise when either the identity is mapped to coherent terms or vice versa.
To avoid these issues, we define the unitarity of a noise channel to be the average purity of output states, with the identity components subtracted, averaged over all pure states. That is, we define (4) where the normalization factor is chosen so that u(I) = 1 and E is defined so that E (A) = E(A) for all traceless A and E (1 d ) = 0. Equivalently, if {A 2 , . . . , A d 2 } is any set of traceless and trace-orthonormal operators (e.g., the Paulis), then we can define the generalized Bloch vector r(ρ) of a density operator ρ with unit trace to be the vector of d 2 − 1 expansion coefficients Our definition of the unitarity is then equivalent to that is, the average squared length (i.e., Euclidean norm) of the generalized Bloch vector after applying the map E with the component due to the identity subtracted off.

II. THE ESTIMATION PROTOCOL
We now present a protocol for characterizing the unitarity of the noise in an experimental implementation of a unitary 2-design G under the assumption that the experimental implementation of any U ∈ G can be written as U • E where U denotes the channel corresponding to conjugation by U and E is a completely positive, trace-preserving (CPTP) channel independent of U . (Note that, as in all randomized benchmarking papers, the assumption that E is independent of U can be relaxed without dramatically effecting the results [10,13,16].) The protocol is to repeat many independent trials of the following.
• Estimate the expectation value Q j of an operator Q after preparing the state ρ and applying the sequence U j = U jm U jm−1 . . . U j1 of operators; that is, in the ideal case E = I.
We will show in Sec. IV, Theorem 5 that under these assumptions on the noise the expected value of Q 2 j over all random sequences j obeys for trace-preserving noise, where A and B are constants incorporating SPAM and the nonunitality of the noise and u(E) ∈ [0, 1] is the unitarity of the noise defined in Eq. (4), with u(E) = 1 if and only if E is unitary. Therefore estimating E j [Q 2 j ] for multiple values of m using the above protocol and fitting to Eq. (8) gives an efficient and robust estimator of the unitarity.
Note that, as opposed to standard randomized benchmarking, we are considering the expectation of an operator Q rather than the probability of a single outcome. This is chosen to maximize the values of the constants to enhance the signal, though it may introduce overhead in the number of measurements required for a single sequence. Also note that unlike in standard randomized benchmarking, we do not require the unitary 2-design to be a group since we do not require an inverse operation, or even that the set G is closed under composition.
More generally, some experimental noise E may be tracedecreasing with an average incoherent survival probability which is the amount of the trace of the quantum state ψ that survives the error channel E, averaged over the Haar measure dψ. When E is itself the average noise over G, the average incoherent leakage rate can be estimated by where C is a constant determined by SPAM [16]. For trace-decreasing noise, the standard fidelity decay curve in Eq. (8) can be generalized to for some constants A and B where The above protocol is a variation of standard randomized benchmarking experiments, and is very similar to the protocol for estimating incoherent leakage presented in Ref. [16]. In particular, one estimates an exponential decay rate in an exactly analogous manner (see Eq. (8)) and the result is obtained in a manner that is robust to SPAM.
However, there are three small but crucial differences to the protocol presented in Ref. [16]. Firstly, the preparation and measurement procedures in the incoherent leakage protocol of Ref. [16] are ideally the maximally mixed state and the trivial measurement respectively, whereas in the current paper, the preparation should be as close as possible and the measurement is of the expectation value of an operator. Secondly, the current protocol requires a unitary 2-design, whereas the incoherent leakage protocol only requires a unitary 1-design (although it also works for a unitary 2-design). Finally, the post-processing is different, since in the present paper the survival probabilities for the individual sequences are squared before they are averaged.

III. NUMERICAL SIMULATIONS
We illustrate numerical results for our protocol under a variety of single-qubit noise models in Figs. 1 and 2. In both these simulations, we assume there is no SPAM and choose G to be the single-qubit Clifford group.
In Fig. 1 we show two runs. In the first we set E to be some fixed (systematic) unitary chosen randomly according to the Haar measure (Haar-random unitary) and some near-identity unitary represented by a rotation of 0.1 radians around the Xaxis of the Bloch sphere (near-identity unitary).
In Fig. 2, we show different types of unital noise composed with the nonunital channel to simulate relaxation to a ground state. The particular unital channels we consider are a near-identity unitary, a Haarrandom unitary, and gate-dependent noise channels corresponding to choosing a fixed perturbation of the eigenvalues of a unitary g by e i to simulate over/under-rotation errors, where the perturbations are chosen independently and uniformly from [−0.1, 0.1] radians for each gate (rotation channel).
Note that the statistical fluctuations in both figures arise from sampling a small number of random sequences (relative to the total number). A perturbation expansion of the form E = I − rδ (where r is the gate infidelity of E to the identity) together with appropriate bounds on the diamond norm can be used to bound these fluctuations and show that they must decrease with gate infidelity, as in Ref. [13]. However, a more detailed analysis is complicated by the complexity of the relevant representation theory (that is, four-fold tensor products).
Finally, we consider the unitarity of random channels drawn from the random ensemble of Bruzda et al. [18], using the QuTiP software package [19] to draw channels and compute their unitarity 1 . As shown in Figure 3, the distribution of unitarities depends strongly on the Kraus rank of the random channel. Moreover, as demonstrated in Figure 4, this information is correlated with, but distinct from, the average gate fidelity.

IV. DERIVATION OF THE FIT MODELS
We now derive the decay curve in Eq. (11) for trace-nonincreasing noise and show how the decay curve in Eq. (8) emerges as a special case for trace-preserving noise.
Since we are dealing with sequences of channels, it will be convenient to work in the Liouville representation. A quantum channel is a completely positive (CP) linear map E : C d1×d1 → C d2×d2 , although we will only consider dimension-preserving channels (i.e., d 1 = d 2 ). Since a quantum channel is a linear map between finite-dimensional vector 1   spaces, it is always possible to represent it as a matrix acting on basis coefficients in some given bases for the vector spaces.
In order to construct the Liouville representation of channels, let A = {A 1 , . . . , A d 2 } be an orthonormal basis of C d×d according to the Hilbert-Schmidt inner product A, B = TrA † B. Any density matrix ρ can be expanded as ρ = k∈N d 2 A k , ρ A k and so we can identify ρ with a column vector |ρ) ∈ C d 2 whose kth entry is A k , ρ . The Liouville representation of a channel E is then the unique matrix E ∈ C d 2 ×d 2 such that E|ρ) = |E[ρ]), which has entries E kl = A k , E(A l ) = (A k |E|A l ). An immediate consequence of the uniqueness of E is that the composition of abstract maps is represented in the Liouville representation by matrix multiplication.
The Liouville representation of unitary channels forms a unitary projective representation of the unitary group U (d).
When we wish to emphasize the Liouville representation as a formal representation (rep) of the unitary group U (d) (or sub-groups thereof), we will use the notation φ L (U ) instead of U . With this notation, it is easy to verify that φ L is indeed a unitary representation of U (d), since the Liouville representation of composition is matrix multiplication and it can easily be verified that φ L (U † ) = φ L (U ) † .
Any representation φ of a semisimple group G [such as SU (d)] over a vector space V can be unitarily decomposed into a direct sum of irreducible representations (irreps) l φ l ⊗ 1 n l , where the l label the irreps and the n l are the corresponding multiplicities and a rep φ over a vector space V is irreducible if there are no nontrivial subspaces of V that are invariant under the action of φ. A particularly important irrep for this paper is the trivial irrep φ T such that φ T (g) = 1 for all g ∈ G.
In the Liouville representation, vectors b ∈ C d 2 are in oneto-one correspondence with operators B ∈ C d×d , so invariant (vector) subspaces under the Liouville representation can be identified with operator subspaces that are invariant under conjugation in the canonical (i.e., d × d matrix) representation. In particular, the identity operator 1 is invariant under conjugation by any unitary, so |1) is an invariant subspace of the Liouville representation corresponding to a trivial irrep.

We now fix
where ⊕ denotes the matrix direct sum and we refer to φ u (U ) as the unital irrep, which has dimension d 2 − 1. Furthermore, any CP channel E can be written in a corresponding block form as where we refer to E sdl , E n and E u as the state-dependent leakage, nonunital and unital blocks respectively. We now show how E u is related to the definition of the unitarity in Eq. (4).

Proposition 1. The unitarity of a channel E is
Proof. For any operator A, TrA † A = (A|A) and E = P u E where P u is the projector onto the unital irrep, so Eq. (4) can be rewritten as where O = dψ|ψ)(ψ|, and the second line follows from the identity P u |ψ) = |ψ − 1 d /d) and the slight abuse of notation Since O commutes with the action of the unitary group, Schur's lemma implies that it is a weighted sum of projectors onto the irreps of φ L , The projector onto the trivial irrep is P T = |A 1 )(A 1 | and so Because TrO = 1 from the normalization of the Haar measure, we can solve for λ u in the expression and we find λ u = 1/d(d + 1). Plugging this in and using P u P T = 0 gives the final result. Before we derive the decay curve in Eq. (11) using the expression for the unitarity from Proposition 1, let us first simplify the quantity of interest. The expectation value of Q given that the sequence j was applied is where ρ is the experimental state preparation. Noting that [20] the expected average of the squares is where U ⊗2 avg = |G| −1 g∈G g ⊗2 , we define the averaged operator M = U ⊗2 avg E ⊗2 U ⊗2 avg , and we have used the fact that |G| −1 g∈G φ(g) is the projector onto the trivial subreps for any rep φ of a group G [21]. Thus, to derive the fit model we must first identify the trivial irreps of G in φ L (U ) ⊗2 , since this is where M is supported.
avg is supported on a two-dimensional subspace spanned by |1 d 2 ) and |S), where S is the SWAP operator.
Proof. Define χ R (g) = TrR(g) as the character of the rep R. Then we can use Schur's orthogonality relations to count the number of trivial irreps. Let χ R , χ R = |G| −1 g∈G χ * R (g)χ R (g) denote the character inner product for G. From the direct sum structure in Eq. (14), the number of trivial irreps is Since χ u is real-valued, we have χ 1 , χ 2 u = χ u , χ u . If G acts irreducibly on the unital block [22], then χ u , χ u = 1 and the number of trivial irreps is 2.
The two trivial irreps in φ ⊗2 L are spanned by the orthonormal vectors |B 1 ) and |B 2 ) where and S is the SWAP operator. To check this, note that since both identity and SWAP are invariant under conjugation by U ⊗ U and S 2 = 1 d 2 . Since φ L (U ) ⊗2 is a unitary rep, B 1 and B 2 are the first two elements of a two-qudit orthonormal Schur basis {B j } for φ ⊗2 L and so correspond to trivial irreps. Therefore M is zero except for the 2 × 2 submatrix supported on |B 1 ) and |B 2 ). These vectors have the same span as |1 d 2 ) and |S).
The next proposition characterizes the averaged operator on the supported subspace.
Proof. We will establish the matrix elements with respect to |1 d 2 ) and |S), and the claims about the B i basis will follow by taking appropriate linear combinations. Because the B i basis is invariant, we can ignore the average unitary terms in M. We first find that Next we can use the identity S, A ⊗ B = Tr S(A ⊗ B) = Tr(AB) and the fact that E(A † ) = E(A) † to find where · F denotes the Frobenius norm. The expression for (1 d 2 |E ⊗2 |S) follows similarly using the adjoint channel. Finally, we can use the expansion S = k A k ⊗ A † k for any orthonormal operator basis A k to obtain The claim in the proposition is established by using the form of Eq. (15) and the definition of the B i basis from Eq. (24) and taking various linear combinations. We omit these tedious details.
The final step in deriving the fit model is to analyze the eigenvalues and eigenvectors of the averaged operator. Proof. Since the averaged operator vanishes almost everywhere, we only need to consider the 2 × 2 submatrix derived above. The nontrivial eigenvalues are This spectrum is degenerate precisely when the terms under the square root both vanish (since both terms are nonnegative). Whenever the spectrum is nondegenerate, there are trivially two distinct eigenvectors, so we only need to deal with the degenerate case. We will break the analysis for the degenerate spectrum into two nontrivial cases, M 11 = M 22 and either M 12 = 0 or M 21 = 0, exclusive. There are also two trivial cases: when M 12 = M 21 = 0, the matrix M is already diagonal and we are done. We ignore the pathological case when M 11 = 0, since this corresponds physically to a state that is never observable. In both nontrivial cases, we will make use of the two-qudit state Π a = 1−S d(d−1) , the maximally mixed state on the antisymmetric subspace. Expanding this state in the B i basis gives The key feature of this state is that π 2 < 0. Case 1: M 12 = 0. In this case, where λ > 0 and y ≥ 0. Taking the mth power gives If we perform the measurement {Π a , 1 − Π a } on a system prepared in the state 1 d 2 /d 2 which evolves under M m , then the probability of observing the outcome Π a is Since λ, π 1 > 0, y ≥ 0, and π 2 < 0, in order for this to be a probability for all m, we require y = 0 and so M is actually diagonal. Case 2: M 21 = 0. In this case, where λ > 0 and y ≥ 0. Taking the mth power gives Therefore the probability of detecting the system (i.e., measuring 1 d 2 ) when a system is prepared in the state Π a and evolves under M m is Again since λ, π 1 > 0, y ≥ 0, and π 2 < 0, for this to be a valid probability for all m, we require y = 0 and so M is actually diagonal. We now have all the ingredients to derive the fit models of Eqs. (8) and (11).
Theorem 5. For time-and gate-independent noise, the expected value E j [Q 2 j ] obeys the decay equation for trace-preserving noise, and for trace-decreasing noise it obeys where λ ± are given by Eq. (28), λ + + λ − = s 2 inc + u(E), and the constants A and B depend only on state preparation and measurement errors.
Proof. Proposition 4 establishes that the matrix M is diagonalizable by a similarity transform with eigenvalues given by Eq. (28). From Eq. (23), we can diagonalize M and absorb the similarity transform into |ρ ⊗2 ) and (Q ⊗2 | as SPAM, yielding Trace-preserving noise is a special case of this, since if E is TP, then by Prop. 3 we have λ + = s 2 inc = 1 and so λ − = u(E).

V. PROPERTIES OF THE UNITARITY
We now prove some properties of the unitarity for CPTP channels that make it a practical quantification of the coherence of a channel. We begin by proving that the unitarity and the average incoherent survival probability can be used to bound the nonunital and state-dependent leakage terms which are subtracted off in the definition of unitarity in Eq. (4).

Proposition 6. For any channel E,
Proof. Consider the maximally mixed states on the symmetric and antisymmetric subspaces, respectively. Expanding these states in the B i basis gives Preparing the state Π s (Π a ), evolving under M and then mea- respectively, where we have used Proposition 3. Since both these expressions are probabilities and so are both greater than or equal to zero, adding the two inequalities and solving for E n 2 + E sdl 2 gives Eq. (36). Furthermore, if the noise is trace-preserving, then Proof. The unitary invariance u(V • E • U) = u(E) follows immediately from the invariance of the trace under cyclic permutations.
Since the norms of vectors are always nonnegative, u(E) = 1 only if E is trace-preserving and unital by Eq. (36), in which case the adjoint channel E † is also a channel [23] and so the eigenvalues of E † E (i.e., the singular values of E) are all bounded by one [24]. Therefore u(E) = 1 only if E is unital and all the eigenvalues of E have unit modulus and consequently if |det E| = 1. However, the only channels with |det E| = 1 are unitary channels [25]. Since u(E) is unitarily invariant and u(I) = 1, u(E) = 1 if and only if E is unitary, as claimed.
We now show that the unitarity can be used with the gate infidelity to quantify how far a channel is from a depolarizing channel. Proof. Any channel with infidelity r to the identity can be written as E = I − r∆ where the diagonal entries of ∆ are nonnegative and Tr∆ = d(d + 1). We then have with equality if and only if E u is diagonal. The term k ∆ 2 kk is uniquely minimized for nonnegative ∆ kk subject to the constraint Tr∆ = d(d + 1) by setting ∆ kk = d/(d − 1) (that is, by setting all the diagonal entries to be equal).
Finally, we give a simple example that shows that the unitarity is not a monotone, in the sense that it can oscillate under composition of channels. Consider the two (nearly) dual channels, E 0 (ρ) = Tr(ρ)|0 0| and Then the unitarity of E 0 and 1 2 E † 0 are both zero, while the unitarity of the composed channel 1 2 E 0 E † 0 is 1/12. We note that for some restricted classes of channels the unitarity is indeed a monotone. For example, a trivial application of von Neumann's trace inequality shows that if the singular values of the unital block are all less than or equal to 1 (which holds for all qubit channels and all unital channels), then it is a monotone for trace-preserving channels.

VI. CONCLUSION
In this paper, we have shown that the coherence of a noise source can be quantified by the unitarity, which corresponds to the change in the purity (with the identity components subtracted off) averaged over pure states. We have presented a protocol for efficiently estimating the unitarity of a generic Markovian noise source that is robust to SPAM errors and illustrated the performance of our protocol through numerical examples.
We have also proven that the unitarity is 1 if and only if the noise source is unitary and provided a tight lower bound for the unitarity in terms of the infidelity (which can be estimated using randomized benchmarking [10]).
Our present results also have direct implications for the incoherent leakage protocol when applied to a unitary 2-design, since the variance over random sequences of fixed length for the protocol in Ref. [16] is which decays faster with m for fixed s inc if the unitarity is smaller (and hence the two decay rates in the fit curve for determining the unitarity, λ ± , are smaller). A lower variance over sequences allows a more precise estimation of the average incoherent survival probability for a fixed number of experiments. Similar implications may also hold for standard randomized benchmarking since u(E) can easily be seen to be one of the eigenvalues of the averaged operator in Ref. [13] that determines the variance and is precisely the eigenvalue that determines the asymptotic variance. However, in order to establish a concrete bound, it would have to be shown that u(E) is in fact the largest eigenvalue. We leave open the problem of finding necessary and sufficient conditions for when u(E) is a monotone, or finding other quantities that are monotonic in general.
Finally, a pressing open problem identified in this paper is to obtain an improved bound on the worst-case error in terms of both the infidelity and the unitarity. Such a bound, in conjunction with standard randomized benchmarking and the protocol presented in this paper, would substantially reduce the effort required to certify that an experimental implementation is near (or even below) the threshold for fault-tolerant quantum computation.

ACKNOWLEDGMENTS
This work was supported by the ARC via EQuS project number CE11001013, by IARPA via the MQCO program, and by the US Army Research Office grant numbers W911NF-14-1-0098 and W911NF-14-1-0103. STF also acknowledges support from an ARC Future Fellowship FT130101744.