Quantum error correction decoheres noise
Typical studies of quantum error correction assume probabilistic Pauli noise, largely because it is relatively easy to analyze and simulate. Consequently, the effective logical noise due to physically realistic coherent errors is relatively unknown. Here we prove that encoding a system in a stabilizer code and measuring error syndromes decoheres errors, that is, converts coherent errors to probabilistic Pauli errors, even when no recovery operations are applied. Two practical consequences are that the error rate in a logical circuit is well-quantified by the average gate fidelity at the logical level and that essentially optimal recovery operators can be determined by independently optimizing the logical fidelity of the effective noise per syndrome.
Quantum computers are likely to dramatically outperform classical computers, provided that errors can be corrected enough to make the output reliable. Errors in a quantum computer can take many forms with differing impacts on a error-correction procedure. Most studies of the performance of quantum error-correcting codes only consider probabilistic Pauli errors because they are easy to simulate via the Gottesman-Knill theorem Aaronson and Gottesman (2004). However, in real systems, it is likely that other noise will also be present.
Determining the performance of an error-correcting code at the logical level under general noise is complicated because such noise is harder to simulate. Previous approaches have expanded the class of errors to some larger class that can still be efficiently simulated Gutiérrez and Brown (2015), performed full density-matrix simulations Gutiérrez et al. (2016), used tensor network descriptions of specific codes Darmawan and Poulin (2017); Bravyi et al. (2017) or effective logical process matrices Rahn et al. (2002); Fern et al. (2006); Chamberland et al. (2017). These methods are sub-optimal because they either require a huge amount of resources to simulate or are indirect approximations. They also do not easily give structural insight because extrapolating the effective logical noise from the description of the encoded state is difficult and determining the scaling with parameters of interest typically requires extensive recalculations.
Optimistically, one may hope that a (numerical or analytical) estimate of the infidelity of the logical noise under a probabilistic Pauli channel generalizes directly to general logical noise. However, even quantifying the error becomes more complicated for more general noise. The “error rate” due to a noise process acting on a -level system is often experimentally quantified via the average gate infidelity to the identity (hereafter the infidelity)
because it can be efficiently estimated via randomized benchmarking Emerson et al. (2005, 2007); Dankert et al. (2009); Knill et al. (2008); Magesan et al. (2011). However, theoreticians often report rigorous bounds on the performance of a quantum error-correcting code or a circuit in terms of the diamond distance to the identity (hereafter the diamond distance) Kitaev (1997)
where and the maximization is over all -dimensional pure states (to account for the error introduced when acting on entangled states).
which scale optimally with respect to and Sanders et al. (2015). For unitary noise, scales as , though it does not necessarily saturate the upper bound of eq. 3; this scaling follows from the magnitude of the coherent (non-Pauli) part of the noise Kueng et al. (). Pauli noise saturates the lower bound of eq. 3 and the effect of coherent noise is often assumed to be negligible, so that experimental infidelities are often compared to diamond distance targets to determine whether fault-tolerance is possible Sanders et al. (2015). However, even if coherent errors make a negligible contribution to the infidelity, they can dominate the diamond norm Wallman and Emerson (2016). Because of this uncertainty about how to quantify errors effectively, it is unclear what figure of merit recovery operations should optimize and how to quantify the logical error rate Gutiérrez et al. (2016); Chamberland et al. (2017); Iyer and Poulin ().
Previous studies have shown that the contribution to the logical noise from the coherent part of the physical noise decays exponentially as a function of code distance Fern et al. (2006), although the decay rate was only given as an abstract property of the noise map. Recently, the decay rate was analyzed for specific noise models in the repetition code Greenbaum and Dutton (2018).
In this paper, we directly relate the decay rate of coherent terms at the logical level of a general stabilizer code to the infidelity of the physical noise of a general local noise process, which can be estimated by randomized benchmarking. Further, we give physical motivation for the decoherence of errors with increasing code distance by relating the scaling of errors to projective syndrome measurements. We demonstrate that—even without applying recovery operations—encoding a system in a quantum error correcting code and measuring error syndromes decoheres errors, that is, reduces them to probabilistic Pauli errors. To isolate the contribution from local noise, we assume that there is no other contributing noise. That is, encoding, syndrome measurements, recovery operations, and decoding are all assumed to be noiseless.
Our results show that the effective logical noise is well-characterized by the logical infidelity. This provides a rigorous justification for choosing recovery maps to independently optimize the logical fidelity per syndrome (instead of, for example, optimizing the diamond norm of the logical noise averaged over all syndromes). Complementary results on the scaling of the diamond distance with quantum error correction protocols were independently obtained in ref. Huang et al. ().
The paper is structured as follows. We first introduce Markovian noise processes and review the process matrix formalism, a convenient representation of quantum channels (not to be confused with the matrix representation). We then give an expression for the infidelity in terms of this representation and discuss the implications and bounds on the entries of a process matrix in terms of its infidelity. Next, we introduce stabilizer codes and, using the aforementioned bounds, discuss the behavior of the effective logical noise of an encoded state after syndrome measurements with and without the application of recovery operations in terms of the physical infidelity of the qubits. We conclude by discussing some implications of our work and discuss how our results relate to existing results showing coherent errors at the logical level.
Ii Markovian Noise Processes
We represent quantum states and measurements of a -dimensional system by vectors as follows. Let be the canonical basis of and be an arbitrary trace-orthonormal basis of respectively, that is, for all . We will generally choose to be the set of normalized (physical or logical) Pauli operators, , or tensor products thereof. We define a map by setting for all and extending to a linear map, so that
Defining , we have
A Markovian noise process is a linear map that maps valid quantum states of one system to valid quantum states of another system, and so is completely positive and trace-preserving (CPTP). Let and be trace-orthonormal bases for the input and output systems respectively. Then
where we abuse notation slightly by using to denote both an abstract map and its matrix representation . Note that is a state of the output system and so is expanded relative to via eq. 4. The composition of two channels is then given by the standard matrix product of the process matrices.
The average infidelity of a single-qubit noise process with the identity in terms of process matrices is Kimmel et al. (2014)
The infidelity only captures the effects of the Pauli part of the noise, that is, the diagonal part, whereas the disconnect between the infidelity and the diamond norm in eq. 3 for non-Pauli noise is due to the off-diagonal terms.
Setting and defining the single-qubit error matrix , we have the following bounds on the matrix entries of in terms of the infidelity.
For any single-qubit Markovian noise process with infidelity , \cref@addtoresetequationparentequation
for all .
where denotes the channel that acts via conjugation by , is a valid channel whose process matrix is the diagonal part of whose singular values are consequently the diagonal entries. We can then write Ruskai et al. (2002) where the must satisfy
Equation 8d holds as the Euclidean norm of any column of is upper-bounded by 1 where is the unital block obtained by deleting the first row and column of Ruskai et al. (2002). Note: The term in the square root was only kept to ; an term was dropped, reducing the inequality from . This convention will be followed for the remainder of the paper. This bound can be tightened further by considering unitarity Wallman et al. (2015). ∎
Iii Stabilizer Codes
We now review stabilizer codes; for more details, see, for example, Ref. Gottesman (2010). Let and . An -qubit Pauli operator is the tensor product of single-qubit Pauli operators, and the weight of a Pauli operator is the number of qubits acts on nontrivially. An stabilizer code encodes logical qubits in physical qubits and is distance ; it is defined by an Abelian group of -qubit Pauli operators, which can be described by a set of generators . We can define a set of mutually orthogonal projectors
where is the th entry of the syndrome, , and the code space is the support of . An error is detectable if it maps the support of outside of and has no effect if it acts trivially on , that is, if it is in . The distance of the code is the minimal Pauli weight of an undetectable error that acts nontrivially on . For each error syndrome we can find a Pauli operator satisfying which corrects the error.
We can find a set of operators such that for all and ,
Let be the projective group generated by . Then is a trace-orthonormal set of operators that span the code space. Therefore any operator in the code space can be written as
Iv Effective Noise Under Error Correction
We now prove that, even with bad decoders (or no correction), encoding in an error correcting code decoheres local errors.
For ideal encoding and correction operations, preparing an initial state in the code space, applying a general local -qubit noise process , and performing a syndrome measurement with the outcome maps the system from the support of to that of . Let be the probability of observing the syndrome , which will generally depend upon the input state. Then by section II the effective noise map from to is
where the factor of comes from the normalization of Rahn et al. (2002). Note that it is conventional to apply a “pure error” Poulin (2006) to map back to the code space. We omit this step to highlight the fact that syndrome measurements alone decohere the noise.
For any stabilizer code, the average off-diagonal elements of the logical noise under a local noise process scales as
where is the sign of in the expansion of eq. 11 and we have used from the normalization of the Pauli operators. By the definition of the code distance, and differ on at least qubits for , and . Therefore for any , each term on the right-hand-side of section IV is in by creftype 1 after syndrome measurements. Averaging over the syndromes cancels the in the denominator. ∎
Intuitively, syndrome measurements decohere errors because the act of measuring projects out any Pauli in the expansion of the output state that is not of the form , thus removing the components of the output state corresponding to the additional Pauli operators introduced by coherent noise.
In creftype 2, we proved that any errors are suppressed exponentially with the code distance. In order to conclude that the noise is decohered, we need to show that the off-diagonals of the logical error matrix, , do not scale as the square root of the diagonals. To see that this holds, at least for typical noise in non-degenerate stabilizer codes, note that section IV is linear in . Writing where is an error that only acts nontrivially on qubits in and ,
For a non-degenerate distance stabilizer code, there exists some set of at most qubits such that cannot be corrected, that is, cancelled out when averaged over syndromes. This set contributes a term . By reducing the generators so that at most one generator acts nontrivially as on each for each , we can find some stabilizer such that for all . Let
which will be for typical noise. Then contributes a term that scales as at least to the effective logical error and so the logical infidelity scales as or worse, so that the off-diagonals are, at worst, proportional to the diagonals of the logical error matrix.
As increases, the scaling described above causes the effective logical noise to become progressively less coherent so that the Pauli twirl approximation captures the logical noise more effectively. However, due to contributions from the coherent part of the physical noise to the Pauli part of the logical noise, approximating the physical noise as Pauli in order to calculate the logical noise produces inaccurate results as observed previously Gutiérrez et al. (2016); Greenbaum and Dutton (2018).
In this paper, we have shown that for generic local noise, coherent errors are decohered by syndrome measurements in error correcting stabilizer codes. Consequently, error rates in logical circuits are well-quantified by the logical infidelity. Therefore it is appropriate to choose recovery operators to optimize the logical fidelity, instead of other measures such as the diamond norm. This dramatically simplifies the process of selecting recovery operators for general noise because the fidelity is a linear function of quantum channels and so we can optimize the fidelity of the logical noise for each syndrome independently, as noted in Chamberland et al. (2017). By contrast, if we tried to optimize the diamond norm of the average logical noise, we would have to simultaneously optimize all recovery operators.
While we have only explicitly considered independent errors, note that our arguments apply directly to correlated errors of the form
by linearity. The only nontrivial issue is identifying a scaling parameter akin to the single-qubit infidelity.
Previous results have demonstrated significant logical coherent errors Fern et al. (2006); Gutiérrez et al. (2016), namely, off-diagonals that scale as compared to diagonals that scale as . However, these results were all for distance 3 codes and are consistent with our results as for such codes, giving diagonals that scale as and off-diagonals that scale as by creftype 2. Numerically, significant discrepancies between the logical diamond norm error with and without Pauli twirling (which removes the coherent part of the noise) at the physical level have been observed for high distance surface codes Darmawan and Poulin (2017) (up to distance 10). These discrepancies have been interpreted as suggesting significant logical coherent errors Greenbaum and Dutton (2018). Our results show that these discrepancies are almost entirely due to contributions to the logical fidelity from the coherent part of the noise, though for a specific syndrome and noise model, the effective logical noise may appear coherent. That is, the effective logical noise is generically very close to a Pauli channel on average, however, it may not be the Pauli channel one would predict from the Pauli twirl of the physical noise.
This research was supported by the Canadian federal and Ontario provincial governments through an NSERC CGS-M and an Ontario Graduate Scholarship. This research was undertaken thanks in part to funding from TQT, CIFAR, the Government of Ontario, and the Government of Canada through CFREF, NSERC and Industry Canada. MG and KRB were supported by the ODNI-IARPA LogiQ program.
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