Optimal arbitrarily accurate composite pulse sequences
Abstract
Implementing a single qubit unitary is often hampered by imperfect control. Systematic amplitude errors , caused by incorrect duration or strength of a pulse, are an especially common problem. But a sequence of imperfect pulses can provide a better implementation of a desired operation, as compared to a single primitive pulse. We find optimal pulse sequences consisting of primitive or rotations that suppress such errors to arbitrary order on arbitrary initial states. Optimality is demonstrated by proving an lower bound and saturating it with solutions. Closedform solutions for arbitrary rotation angles are given for . Perturbative solutions for any are proven for small angles, while arbitrary angle solutions are obtained by analytic continuation up to . The derivation proceeds by a novel algebraic and nonrecursive approach, in which finding amplitude error correcting sequences can be reduced to solving polynomial equations.
pacs:
03.67.Pp, 82.56.JnI Introduction
Quantum computers are poised to solve a class of technologically relevant problems intractable on classical machines Nielsen and Chuang (2004), but scalable implementations managing a useful number of qubits are directly impeded by two general classes of errors Merrill and Brown (2012). On one hand, unwanted systembath interactions in open quantum systems lead to decoherence and, on the other, imperfect controls for addressing and manipulating qubit states result in cumulative errors that eventually render large computations useless.
Systematic amplitude errors, the consistent over or under rotation of a singlequbit unitary operation by a small factor , are one common control fault. The discovery of a protocol for the complete and efficient suppression of these errors would greatly advance the field of quantum control with applications as far ranging as implementing faulttolerant quantum computation and improving nuclear magnetic resonance spectra acquisition. Due to the broad scope of systematic amplitude errors, this problem has been attacked repeatedly by a variety of methods with varying degrees of success Tycko et al. (1985); Levitt (1986); Brown et al. (2004); Ichikawa et al. (2012); Jones (2013); Husain et al. (2013). A concept common to most approaches is the composite pulse sequence, in which some number of carefully chosen erroneous primitive unitary operations, or pulses, are applied successively such that a target ideal rotation is approximated to some order with an exponentially reduced error .
In the realm of quantum computation, the criteria for useful pulse sequences are stringent: (1) For each order , a procedure for constructing a pulse sequence correcting to that order is known. (2) This construction gives sequence lengths that scale efficiently with , that is with as small as possible. (3) Sequences should be ‘fullycompensating’ or ‘Class A’ Levitt (1986), meaning they operate successfully on arbitrary and unknown states (in contrast to ‘Class B’ sequences that only operate successfully on select initial states). (4) Although finite sets of universal quantum gates exist Nielsen and Chuang (2004), ideally sequences should be capable of implementing arbitrary rotations so that quantum algorithms can be simplified conceptually and practically.
One finds that there are currently no sequences satisfying all four of these criteria and suppressing systematic amplitude errors. In the literature, SCROFULOUS Cummins et al. (2003), PB, BB Wimperis (1994) satisfy criteria (3) and (4) but offer corrections only up to order . Unfortunately, generalizations of these to arbitrary and come with prohibitively long sequence lengths, so that criterion (2) ends up unsatisfied. Typically, a sequence correct to order is recursively constructed from those at order , resulting in an inefficient sequence length Brown et al. (2004), although numerical studies suggest that efficient sequences with exist Brown et al. (2004). To date, other classes of systematic control errors Viola et al. (1999); Khodjasteh et al. (2010); Souza (2012) do not fare better.
There are some provable successes in efficient pulse sequences, though. However, to find them, one must relax criterion (4) that requires arbitrary rotations. For example, if one restricts attention to correcting rotations in the presence of amplitude errors, Jones proved the impressive result that sequences with Jones (2013); Tycko et al. (1985) are possible. Uhrig efficiently implements the identity operator in the presence of dephasing errors with Uhrig (2007). If we also relax the criterion (3) and settle for specialized Class B sequences that take to (those we call inverting sequences), Vitanov has found efficient narrowband sequences for amplitude errors also with Vitanov (2011). Notably, both Uhrig’s and Vitanov’s results were achieved via algebraic, nonrecursive processes. In fact, as we show, a more generalized algebraic approach in the amplitude error case can reinstate the crucial criteria (3) and (4), while maintaining Vitanov’s efficient length scaling.
Our main result is exactly such an algebraic generalization, a nonrecursive formalism for systematic amplitude errors. With this, we prove a lower bound of for Class A sequences comprised of either primitive or rotations, then constructively saturate this bound to a constant factor with (plus a single initializing rotation). The improvement of these new sequences over prior stateoftheart is illustrated in Table 1. We derive optimal closedform solutions up to for arbitrary target angles, and perturbative solutions for any , valid for small target angles. We then analytically continue these perturbative solutions arbitrary angles up to . Since any random or uncorrected systematic errors in the primitive pulses accumulate linearly with sequence length, optimally short sequences such as ours minimize the effect of such errors.
We define the problem statement for amplitudeerror correcting pulse sequences mathematically in Section II, leading, in Section III, to a set of constraint equations that such pulse sequences must satisfy, which is then solved in Section IV by three approaches: analytical, perturbative, and numerical. The analytical method is interesting as it gives closed form solutions for low order sequences in a systematic fashion. The perturbative method relies on invertibility of the Jacobian of the constraints and is used for proving the existence of solutions for select target angles. The numerical method is the most straightforward and practical for higher orders, giving optimally short pulse sequences for correction orders up to . Section V then presents several generalizations of our results, including discussions on narrowband toggling, nonlinear amplitude errors, random errors, and simultaneous correction of offresonance errors. Finally, we point out differences and similarities between our sequences and existing art in Section VI, and conclude in Section VII.
Ii Pulse sequences
A single qubit rotation of target angle about the axis is the unitary , where is the vector of Pauli operators. Without affecting the asymptotic efficiency of our sequences, Euler angles allow us to choose , and consequently we define for . However, we only have access to imperfect rotations that overshoot a desired angle by , . With these primitive elements, we construct a pulse sequence consisting of faulty pulses:
(1) 
Denote by the vector of phase angles , which are our free parameters. Leaving each amplitude as a free parameter (e.g. SCROFULOUS Cummins et al. (2003)) may help reduce sequence length, but we find that a fixed value leads to the most compelling results.
The goal is to implement a target rotation (or, without loss of generality, by the replacement ) including the correct global phase, with a small error. The trace distance Nielsen and Chuang (2004)
(2) 
is a natural metric for defining errors between two operators , Brown et al. (2004). We demand that the pulse sequence implements
(3) 
so that the corrected rotation has trace distance with the same small leading error . Thus constructed, implements over a very wide range of due to its first derivatives vanishing and so has broadband characteristics Merrill and Brown (2012).
For completeness, we mention other error quantifiers. First, is the fidelity , which is not truly a distance metric, but can be easier to compute and bounds Nielsen and Chuang (2004). The infidelity of is then , which is a commonly used quantifier Merrill and Brown (2012); Jones (2013). Finally, for the specialized Class B sequences called inverting sequences the transition probability is a viable quantity for comparison Vitanov (2011).
Iii Constraint equations
We now proceed to derive a set of equations, or constraints, on the phase angles that will yield broadband correction. We begin very generally in the first subsection by assuming just as mentioned before, but then we specialize in the subsequent two subsections to the case and the case of symmetric sequences, both of which greatly enhance tractability of the problem.
iii.1 Equal amplitude base pulses
To begin, we obtain an algebraic expression for by a direct expansion of a length sequence. Defining ,
(4)  
where indices in the matrix product ascend from left to right, , and are noncommutative elementary symmetric functions generated by Gelfand et al. (1995). The are hard to work with so by applying the Pauli matrix identity
(5) 
we obtain a more useful expression as functions of the phase angles :
(6)  
(7) 
By defining the terminal case , the phase sums are efficiently computable at numeric values of the phases by the recursion using dynamic programming (i.e. start from the terminal case and fill in the table for all desired and ).
Name  Length  Notes 
SCROFULOUS  , nonuniform Cummins et al. (2003)  
P, B  Closedform Brown et al. (2004)  
SK  , numerical Brown et al. (2004)  
, conjectured  
AP (PD)  , closedform  
, analytic continuation  
, conjectured  
ToP  arbitrary , perturbative 
Combining the expansion of with Eq. (3) then imposes a set of real constraints on to be satisfied by any order , length sequence. is obtained by first matching coefficients of the trace orthogonal Pauli operators on either side of Eq. (3). We then obtain in terms of normalized error and normalized target angle the necessary and sufficient conditions
(8) 
Second, the complex coefficients of are matched, giving complex equations linear in the phase sums , or real constraints.
However, these constraints are intractable to direct solution, and a simplifying assumption is necessary. It should be reasonable to suspect that the small rotation can be generated by small pure error terms . We will therefore set Brown et al. (2004). Note that is also a tractable case but is related to the pulse case by phase toggling and so need not be considered separately. We give more detail on toggling in Section V.
iii.2 Assuming base pulses of
We now enumerate several key results, due simply to imposing , that apply to all order , length , pulse sequences. First, Eq. (8) reduces to
(9) 
by summing its even and odd parts, justified by noting hence occurs only in even (odd) powers for even (odd). By matching coefficients of powers of , this represents real constraints. Second, the terms in Eq. (9) match if and only if is even. Assuming this, constraints remain. Third, we arrive at our most important result by transforming Eq. (9) with the substitution . This eliminates trigonometric and exponential functions from Eq. (9), and (assuming is even) leaves
(10) 
Upon rearrangement, this is a generating equation for values that the phase sums must satisfy. The functions generated by are, in fact, real polynomials in of degree which generalize those of MittagLeffler Bateman (1940). We can now write
(11)  
(12) 
Eq. (11) is, in our opinion, the simplest and most useful representation of the nonlinear (in ) constraints that form the basis for our solutions.
In our notation the leading error of an order , even , pulse sequence has a simple form,
(13) 
Now, we recognize the operator on the right of Eq. (13) must be unitary. Thus, if a set satisfies Eq. (11) for for any even integer , follows automatically. So we define , the set of constraints resulting from applying to , to consist of the complex equations from Eq. (11) ignoring the real parts for even .
(14) 
Thus, .
In fact, it is not difficult to place a lower bound on the pulse length for a sequence correcting to order using the framework we have so far. This is the first bound of its kind, and, given our solutions of the constraints to come in section IV, it must be tight to a constant factor. Begin the argument by way of contradiction, letting . In examining , observe for , but is a real polynomial in of degree . Hence cannot be satisfied for arbitrary . Likewise, if , then cannot be satisfied for arbitrary . Thus is necessary.
iii.3 Assuming phase angle symmetries
Some constraints in can be automatically satisfied if appropriate symmetries on the phase angle are imposed. A symmetry property of the phase sums, with reversed phase angles , motivates us to impose a palindromic (antipalindromic) symmetry on the phases, () so that for even (odd) . Removing these equations from , we are left with the subset (). By definition, and . In both cases, we have real constraints to be satisfied by real variables or . With what minimum is this possible, and is it of the linear length scaling suggested by our lower bound?
Iv Solving the constraints
We now satisfy the constraints and with sequences of length exactly , using three different methods – analytical, perturbative, and numerical – and achieving the linear lower bound for pulse sequences. Our solutions are nonrecursive; a lower order sequence never appears as part of an order sequence. Table 1 summarizes our results labeled by PD (AP) for the palindromes (antipalindromes), as well as “Tower of Power” (ToP) sequences, a name inspired by their visual appearance in Fig. 1, which are special AP sequences essential to our perturbative proof for that lengthoptimal arbitrary corrections for nontrivial exist.
Subsections IV.1, IV.2, and IV.3 detail respectively the analytical, perturbative, and numerical solution methods and the corresponding results.
iv.1 Closed form solutions
We obtain closedform solutions to and for , presented in Table 2, by the method of Gröbner bases Cox et al. (2007), which we describe here. The sequences AP, AP, and PD are original whereas AP and PD recover SK and PB of Brown et al. Brown et al. (2004) and Wimperis Wimperis (1994) respectively.
Sequence  n  Phase angle solutions 

AP  1  
AP  2  
PD  2  
NS  2  
AP  3  
PD  4 
The key insight in solving the transcendental constraints and with as a free parameter is that any is equivalent to systems of multivariate polynomial equations , for which powerful algorithmic methods of solution are known. This equivalence can be seen by introducing the Weierstrass substitution . Any with variables that has a finite number of zeroes is zerodimensional and has solutions that can always be represented in the form of a regular chain, that is, a finite triangular system of polynomials obtained by taking appropriate linear combinations of elements of . Regular chains are easy to solve as the first equation is a univariate polynomial in whose zeroes can then be substituted into , thus converting it into a univariate polynomial in . Through recursive substitution, all can obtained in a straightforward manner.
Divining these appropriate linear combinations appears to be a formidable task, but surprisingly, they can be deterministically computed by applying algorithms such as Buchberger’s algorithm Buchberger (2006) for computing the Gröbner basis Cox et al. (2007) of . The basis is another system of polynomial equations that shares the same zeroes as , in addition to certain desirable algebraic properties. For example, can readily decide the existence, number of, and location of complex zeroes Buchberger (1998), and by choosing a lexicographic term order, is itself a regular chain Cox et al. (2007). The algorithm generalizes Gaussian elimination for systems of linear equations and finding the greatest common divisor of univariate polynomial equations to systems multivariate polynomial equations: the reader is referred to excellent resources for more information Sturmfels (2005, 2002); Cox et al. (2007). In the Appendix, we also present a brief overview of Gröbner bases and Buchberger’s algorithm for calculating them, including handworked examples for AP and PD, the results of which are part of Table 2.
The regular chains for the remaining sequences AP, AP, and PD solved in Table 2 can be computed by optimized variants of Buchberger’s algorithm Cox et al. (2007) in Mathematica. In each case, closedform is achieved since is a univariate polynomial of at most quartic degree in , and the remaining variables are then given as functions of only . Only the real solutions, which exist for , are physically meaningful. The utility of Gröbner bases for short sequences is clear as is it highly unlikely that these solutions could have been arrived at by hand.
As a curiosity, we also present in Table 2 a closedform solution for , where no symmetry has been applied to the four pulse sequence, denoted NS. NS has one free parameter in the phase angles, which we arbitrarily choose to be . By fixing and solving for the remaining phase angles, one finds that NS continuously deforms between PD and AP and hence generalizes them.
Could one solve for arbitrary by this method? Any arbitrary system of multivariate polynomials is guaranteed to have a Gröbner basis that can always computed in a finite number of steps by Buchberger’s algorithm Buchberger (2006). Thus, complex solutions to zerodimensional with the same number of equations as free parameters can always be found by this method in principle. However, the worstcase time complexity of computing for a system of variables and total degree scales as Dube (1990) and rapidly becomes infeasible. Of greater concern, there is no guarantee that such solutions are real, representing physical phases .
We now prove that there exists real solutions to over a continuous range of for arbitrary , and show how this leads to an efficient constructive procedure for computing arbitrary angle sequences.
iv.2 Perturbative solutions
We may solve and perturbatively. A wellknown theorem of square Jacobian matrices states that any arbitrary function is locally invertible, or analytical, in the neighbourhood about some point if and only if the determinant of its Jacobian matrix is nonzero. Thus, setting , this theorem says that one may always construct a perturbative expansion for over a continuous range about given a valid starting point satisfying if and only if . So long as the Jacobian remains nonzero, one may extend such a solution beyond its neighbourhood by analytic continuation.
However, for arbitrary , what are these valid initial points ? As we do not a priori know of solutions to for arbitrary , such points must be found at some where the problem simplifies. Even then, the problem is nontrivial: for example, imposing phase angle symmetries forces at , but one can readily verify that many such solutions of this form to Eq. 11 suffer from . Using the closedform solutions in Table 2, one finds at that while the Jacobian of the PD sequences is zero, the AP sequences each have a solution with nonzero Jacobian wherein . We now prove that this generalizes to arbitrary , resulting in the special class of antipalindrome sequences with initial values
(15) 
for . Hence, nontrivial real solutions to exist for arbitrary .
iv.2.1 ToP is analytical at
We first transform the function mapping for ToP: This does not affect the magnitude of its Jacobian as for odd due to antipalindromic symmetry, while for even the real part of Eq. 11 is automatically satisfied due to unitarity (see Eq. 13).
The solution to ToP has a simple form . With this solution, a straightforward, if tedious, manipulation of the phase sums shows that elements of Jacobian matrix satisfy the recurrence . The solution to this recurrence is best seen from a combinatorial standpoint. Consider the related puzzle — you begin at the top left corner (1,1) of an checkerboard and would like to reach the position , the lower right corner. You may move only south, southeast, or east at any given time, enforcing the recursion. If, additionally, your first move cannot be south, how many paths exist that achieve your goal? The solution is
(16) 
since you may take any number of southerly steps . If your first move is not restricted, the number of paths is . For later use, define an matrix with the elements .
We will now express in terms of the leftmost column . This is an extension of the path counting problem, in which we may begin our walk to from any leftmost location. Therefore,
(17) 
Now notice that the determinant of does not depend upon the leftmost column. Since and we can always subtract multiples of rows of to obtain . Thus, .
We have reduced the problem to finding the determinant of . We claim that has LUdecomposition
(18) 
This means that is the binomial transform of the Chebyshev triangle . This is proved by looking at the generating functions of and , namely
(19)  
(20) 
These are related by 18). With the LUdecomposition, one can immediately see that , we have . This concludes the proof that ToP sequences exist for a continuous range of small target angles within the neighbourhood of all . , which implies the binomial transform in Eq. (
iv.3 Numerical solutions
Our demonstrations of real, arbitrary angle () solutions for small order () and real, arbitrary order solutions for a continuous range of small angles inspires confidence that real solutions for larger at arbitrary can always be found. Although proving this notion is difficult, the zerothorder analytically continuable solutions provide, in principle, a means of obtaining arbitrary , arbitrary sequences that are exponentially more efficient than a brute force search for solutions to Eq. (11). As long as a sequence for some has a nonzero Jacobian, its phase angles may be continuously deformed into another solution to Eq. (11) in the neighbourhood of .
We present the results of this procedure for ToP and PD to obtain real optimal length solutions over , and provide for the convenience of the reader some solutions derived in this manner at common values of in Table. 3. up to
We provide details of the continuations for ToP and PD in subsections IV.3.1 and IV.3.2. We know more exotic solutions exist, too, and in section IV.3.3 we provide the results of brute force numerical solutions to Eq. (11).
1  AP1  2.09440  1.31607  PD2  1.82348  1.82348  1.16499  AP3  0.74570  2.11099  2.37504  1.10279 

AP2  2.35949  1.35980  1.16499  AP3  2.51806  1.15532  1.66273  1.10279  
AP1  1.82348  0.98400  PD2  1.69612  1.69612  0.88856  AP3  0.87848  1.93555  2.13129  0.93360  
AP2  1.95071  1.44966  1.05957  AP3  2.03611  1.29441  1.64504  1.12057  
AP1  1.69612  0.70433  PD2  1.63334  1.63334  0.69647  AP3  0.98173  1.85668  1.98076  0.77869  
AP2  1.75891  1.50875  0.86557  AP3  1.80090  1.42667  1.61136  0.98560 
PD4  2.26950  1.76948  0.80579  1.93044  1.07009  AP5  2.30757  2.57163  1.03434  0.26267  2.05214  1.05043  

PD4  1.38777  0.59527  2.74476  2.19903  1.07009  AP5  0.44569  1.40804  1.55593  2.45457  2.50831  1.05043 

AP4  1.64767  1.97451  2.92461  0.32956  1.07009  AP5  2.19409  0.13026  2.09113  1.82984  1.72591  1.05043 

AP4  2.62323  0.99049  1.79394  1.52913  1.07009  AP5  2.69800  0.86163  1.92713  1.45034  1.59011  1.05043 
PD4  2.30661  1.30540  0.47998  2.38635  0.88941  AP5  1.86484  2.24227  1.42219  0.43481  1.97474  0.91458  

PD4  1.47070  0.11256  2.96678  1.93782  0.89673  AP5  0.60281  1.44347  1.45031  2.28880  2.29567  0.93241 

AP4  1.05532  2.36238  3.06746  0.26240  0.90343  AP5  1.78432  0.205507  2.21897  1.86132  1.69801  1.05179 

AP4  2.10426  1.11746  1.80109  1.52196  1.15341  AP5  2.17223  0.89078  2.05179  1.39042  1.60052  1.13467 
PD4  2.45079  0.96051  0.28079  2.66423  0.76705  AP5  1.56763  2.19365  1.57024  0.61251  1.94290  0.80141  

PD4  1.51926  0.73603  2.38182  1.76467  0.77110  AP5  0.73268  1.46554  1.41033  2.18996  2.15661  0.82226 

AP4  0.68460  2.64974  3.11442  0.19308  0.77643  AP5  1.62724  0.36022  2.23779  1.88279  1.64971  0.95228 

AP4  1.83302  1.33340  1.70166  1.54125  1.07442  AP5  1.86049  1.22698  1.85066  1.44825  1.59325  1.13458 
PD6  2.48390  1.63561  0.23686  2.03217  2.74686  0.71914  1.03757  AP7  2.13314  1.81689  1.08618 