A unified framework for magic state distillation
and multiqubit gatesynthesis with reduced resource cost
Abstract
The standard approach to faulttolerant quantum computation is to store information in a quantum error correction code, such as the surface code, and process information using a strategy that can be summarized as distillthensynthesize. In the distill step, one performs several rounds of distillation to create highfidelity logical qubits in a magic state. Each such magic state provides one good gate. In the synthesize step, one seeks the optimal decomposition of an algorithm into a sequence of many gates interleaved with Clifford gates. This gatesynthesis problem is well understood for multiqubit gates that do not use any Hadamards. We present an indepth analysis of a unified framework that realises one round of distillation and multiqubit gate synthesis in a single step. We call these synthillation protocols, and show they lead to a large reduction in resource overheads. This is because synthillation can implement a general class of circuits using the same number of states as gate synthesis, yet with the benefit of quadratic error suppression. This general class includes all circuits primarily dominated by controlcontrolZ gates, such as adders and modular exponentiation routines used in Shor’s algorithm. Therefore, synthillation removes the need for a costly round of magic state distillation. We also present several additional results on the multiqubit gatesynthesis problem. We provide an efficient algorithm for synthesizing unitaries with the same worstcase resource scaling as optimal solutions. For the special case of synthesizing controlledunitaries, our techniques are not just efficient but exactly optimal. We observe that the gatesynthesis cost, measured by count, is often strictly subadditive. Numerous explicit applications of our techniques are also presented.
The topological surface code or toric code Dennis et al. (2002) is the most widely known modern approach to quantum error correction. Tolerating noise up to Wang et al. (2003); Raussendorf and Harrington (2007), it has established itself as the frontrunning proposal for quantum computation Jones et al. (2012); Fowler et al. (2012); Nickerson et al. (2014). However, it can not natively support fully universal quantum computation Eastin and Knill (2009). Augmenting the surface code from a static device to a computer requires extra gadgets, which can be realised by a twostep process. In the first step, magic state distillation is used to prepare encoded highfidelity magic states Bravyi and Kitaev (2005). Each of these magic resources provides a faulttolerant gate, also known as a phase gate. In the second step, we decompose any desired unitary into a sequence of gates and Clifford gates, using gatesynthesis techniques to minimise the required number of gates. We paraphrase this paradigm as distillthensynthesize.
After the initial discovery of ReedMuller protocols for magic state distillation Knill (2005); Bravyi and Kitaev (2005), recent years brought several innovations that reduced the cost of magic state distillation. Next came the protocol of Meier. et al Meier et al. (2013), followed by the triorthgonal codes of Bravyi and Haah Bravyi and Haah (2012). The BravyiHaah magic state distillation (BHMSD) protocol converts magic states into magic states with quadratic error suppression, and will be our standard benchmark throughout. Concatenating BHMSD two or three times, will suppress error rates from to between and , which suffices for many near term applications. Once below very small error rates, multilevel distillation Jones (2013a) can further improve distillation yields, though it requires much larger circuits.
Gate synthesis has undergone an even more impressive renaissance, making huge leaps forward since the early days of the SolovayKitaev theorem Kitaev et al. (2002); Dawson and Nielsen (2005). For synthesis of single qubit gates, optimal protocols have been found Kliuchnikov et al. (2013); Ross and Selinger (2016); Bocharov et al. (2015). Here we are primarily interested in the multiqubit gatesynthesis problem Amy et al. (2013); Selinger (2013); Amy et al. (2014); Maslov (2016); Kliuchnikov et al. (2013); Bocharov et al. (2015). For multiqubit circuits generated by CNOT and gates, optimal synthesis is well characterised Amy et al. (2013, 2014); Amy and Mosca (2016); Maslov (2016), though no efficient solver exists for large circuits. This multiqubit gate set requires Hadamards to acquire universality, and so gatesynthesis can be applied to subcircuits separated by Hadamards as shown in Fig. (1a). This class of multiqubit gates is finite and can be exactly synthesized from the relevant gate set. That is, there is no approximation error in this multiqubit synthesis problem and any noise arises from imperfections in the gates used.
However, the anticipated resource cost for faulttolerant quantum computing remains formidable and we seek further reductions. To date, most of this progress came about by treating magic state distillation and gatesynthesis as distinct puzzles. However, one can circumvent the need for subsequent synthesis. As an alternative to inexact synthesis of single qubit rotations, one can prepare special single qubit resources DuclosCianci and Svore (2012); Landahl and Cesare (2013); DuclosCianci and Poulin (2015); Campbell and O’Gorman (2016). In the multiqubit setting, the only known alternative approach prepares the resource state for a Toffoli gate Jones (2013a); Eastin (2013); Paetznick and Reichardt (2013). This work inspired us to ask whether one can directly distill resources for a general class of multiqubit gates.
Here we present a general framework for implementing errorsuppressed multiqubit circuits generated by CNOT and gates. Our approach fuses notions of phase polynomials used in multiqubit gate synthesis Amy and Mosca (2016) with a generalisation of Bravyi and Haah’s triorthogonal matrices Bravyi and Haah (2012). Our work reveals mathematical connections between these concepts, showing our protocols to be formal unifications of previous of gatesynthesis and distillation protocols. For singlequbit smallangle rotations, schemes like Landahl and Cesare (2013); DuclosCianci and Poulin (2015); Campbell and O’Gorman (2016) share some similarity with our current work, insofar as the need for subsequent synthesis is removed. The protocols in Jones (2013a); Eastin (2013) are closer in spirit to our work as multiqubit synthesis for the Toffoli (only) is implicitly performed, but our work makes the connections to synthesis both explicit and general. On a practical level, synthillation is never more expensive than traditional distillthensynthesize. But, for a broad and important class of circuits, synthillation effectively eliminates the need for one round of distillation. For many applications, we need only two or three rounds of BHMSD, so removing one round is a significant advance. Asymptotically, one round of BHMSD uses three raw copies per output, and so by this metric our approach reduces overheads by approximately a third. We emphasise that this resource saving is benchmarked against optimal gatesynthesis, and so is cumulative with resource saving made over naive, suboptimal approaches to gatesynthesis. The synthillation protocol is also compatible with modulechecking O’Gorman and Campbell (2016), which offers further savings in some regimes. We also present several techniques and efficient algorithms for finding gatesynthesis decompositions, which naturally feeds into our synthillation protocol. In general, optimal gatesynthesis appears to be a hard problem, but we make progress by focusing on easy special classes and looking for nearoptimal solutions.
Our first section begins by formalising the exact multiqubit synthesis problem, and outlining our key results. Sec. II presents the synthillation protocol. Sec. III provides the proofs for our gatesynthesis results. Sec. IV goes into a detailed study of several concrete applications. We close with Sec. V, discussing the broader context. All calculations and examples presented here can be reproduced using a Mathematica script in our supplementary material Campbell and Howard (2016a). A more concise account of the synthillation protocol is also available Campbell and Howard (2016b).
We remark that there are several ideas on how to circumvent magic state distillation Bombin and MartinDelgado (2006, 2009); Paetznick and Reichardt (2013); Bombin et al. (2013); Bombín (2015); JochymO’Connor and Laflamme (2014). While these approaches save on the costs associated with magic state distillation, they all incur additional costs that are not immediately apparent. For instance, typically these proposals require extra allocation of resources toward error correction. So far, no alternative has been quantifiably shown to compete with twodimensional topological codes combined with distillthensynthesize. In particular, no alternative has come close to the threshold of the surface code, with current numerics pointing toward 3D gauge colour codes possessing a threshold that is worse by an order of magnitude Brown et al. (2016); Bravyi and Cross (2015). This further motivates expanding the repertoire of techniques within the magic states paradigm.
I Overview
The magic states model was first formalized by Bravyi and Kitaev Bravyi and Kitaev (2005). It assumes certain operations are ideal, free resources. The model is justified because these operations are natively protected against noise in many error correcting codes, including the 2D topological codes such as the surface code and 2D colour codes. The protected operations are called Clifford operations and include: preparation of states, measurement of Paulispin operators (elements of the Pauli group ), unitaries in the Clifford group (denoted , the normalizer of the Pauli group), classical randomness and feedforward. Stabilizer states can be reached from states with Clifford unitaries and also constitute free resources. In contrast, nonstabilizer states and nonClifford unitaries are not natively protected, and so not free from noise and constitute costly resources. To obtain highfidelity nonClifford operations, such as the gate or preparation of magic states, requires several layers of magic state distillation, with each layer comprising many Clifford operations. As such, the cost of magic states is significantly more than a Clifford operation. Throughout we measure resources by counting raw, noisy states consumed. This does not provide the full story as Clifford costs are not entirely negligible Fowler et al. (2013); O’Gorman and Campbell (2016); Maslov (2016), but provides a good starting point for conceiving new protocols. Throughout, we will often refer to a factor 3 saving in costs, and ask the reader to keep in mind that the full resource saving could be much greater than this.
We denote for the subgroup of the Clifford group, which can be implemented with CNOTs and gates, where
(1) 
We define the gate as
(2) 
with . Composing gates in , it was found Amy et al. (2013) that all unitaries in the augmented group can be decomposed as where is some sequence of CNOT gates and belongs to a special class of diagonal unitaries. We define this special class as , with gates in this group having the form
(3) 
where is a computational basis state labelled by a binary string , and is a cubic polynomial of a particular form
(4) 
where , and are linear, quadratic and cubic polynomials. Explicitly,
(5)  
where the coefficients are integers defined modulo 8. Sometimes we will refer to this as a weighted polynomial because the degree terms have coefficients that are weighted by . When it is clear from the context we drop the subscript from , and at times it will be necessary to instead write as the function corresponding to . We will show later that the gates reside in the level of the Clifford hierarchy Gottesman and Chuang (1999a), which explains our choice for the subscript 3. We can directly infer that can be decomposed as where contains only gates, contains only control gates (CS or short) and contains only controlcontrolZ gates (CCZ). All these gates are diagonal in the computational basis with and . We find a special role is played by unitaries composed of CCZ gates, and denote this subgroup as , where the superscript indicates that the associated weighted polynomial has only cubic terms, and so is a homogeneous cubic polynomial. The gate set is not universal, but becomes universal when is promoted to the full Clifford group by including the Hadamard. The strategy of multiqubit gate synthesis is to take a universal circuit and partition it into subcircuits composed from segmented by Hadamards, as illustrated in Fig. (1a). From this one then optimises the decomposition of these subcircuits.
We define the count as following.
Definition 1
For any we define the ancillafree count as
(6) 
It is possible to use fewer gates by exploiting ancilla. Though, to the best of our knowledge, there is not yet a general toolbox for ancillaassisted gatesynthesis and only a few such protocols are known (see e.g. Ref. Jones (2013a); Paetznick and Svore (2014)). In contrast, is well understood and we have techniques for achieving optimality Amy and Mosca (2016). We are interested solely in reducing counts, and do not consider depth or Clifford resources in our assessments of optimality. In Fig. (1b) we show an optimal decomposition for realising a CS gate, and Fig. (1c) shows an optimal decomposition for a combined CS gate and CCZ gate. Individually, a CS gate require 3 gates and a CCZ gate requires 7 gates, but the composite circuit shown calls for only 4 gates where a naive composition of CS and CCZ would have used 10 gates. The benefits of our synthillation protocol will be additional to such smart reductions in gates, and will use many of the same mathematical tools as gatesynthesis.
We find that CCZ gates are more amenable to resource savings than other gates, and so introduce another measure of circuit complexity
Definition 2
For any we define
(7) 
where is the subgroup of composed of CCZ gates.
Clearly, since we can always set and . Furthermore, if then by setting and . However, the decomposition can be more counterintuitive. In Fig. (1d), we show a circuit where contains no CCZ gates, yet the minimisation to find must use a decomposition where both and contain CCZ gates. Having defined and , we can state our main result
Theorem 1 (The synthillation theorem)
Let be a set of unitaries in the family , and . The synthillation protocol can implement with probability and error rate using
(8) 
noisy states of initial error rate , where is a constant in the range .
The constant is bounded and so becomes unimportant in the limit of large circuits. The synthillated need not be implemented in parallel, each unitary maybe injected into a circuit at any point. See Fig. 1a for an example set that are not injected as a tensor product, though the synthillation cost is determined by . It is important to recognise that is error rate on the magic states used rather than a measure of synthesis precision. For inexact synthesis problems, is often used to quantify the precision of an implemented unitary relative to a target unitary. In this context, synthesis is exact. Both our protocol and gatesynthesis Amy et al. (2013); Selinger (2013); Amy et al. (2014); Maslov (2016); Kliuchnikov et al. (2013); Bocharov et al. (2015) will implement a perfect when supplied with perfect magic states. Given imperfect magic states with error , synthillation realises with quadratically suppressed error, whereas using the same magic states gatesynthesis would lead to a implementation of .
Therefore, we instead compare synthillation against distillthensynthesize, which is one round of distillation followed by gatesynthesis. Now both approaches yield error, but have different resource overheads and are summarised in Fig. 2. Asymptotically, our approach is never more expensive than using a round of BHMSD followed by gate synthesis, which would cost ignoring additive constants. Whereas, if synthillation costs the price of using BHMSD with gatesynthesis. This maximum saving is attained whenever as then . This class of circuits is common as quantum algorithms often contain components that consist of classical reversible logic achieved using only Toffoli gates, CNOT gates and NOT gates. For instance, modular exponentiation is simply classical logic and also amounts to the dominant resource cost in Shor’s algorithm Kitaev et al. (2002); Fowler et al. (2004). Furthermore, Toffoli and Hadamard form a universal gate set, so the gate set is universal. Beyond Toffoli circuits, there are many other cases where we obtain close to this saving, which is ensured by the following
Theorem 2
For all acting on qubits, we have . Furthermore, there exists a algorithm for finding both a decomposition (with and ) and also an optimal synthesis of using Clifford+ gates.
We see this theorem at work in Fig. (1d), where a 4 qubit circuit has even though does not contain any CCZ gates. More generally, this shows that scales at most linearly with the number of qubits, whereas Amy and Mosca Amy and Mosca (2016) showed that scales at most quadratically. This quadratic scaling tells us that complex circuits may have which entails . In such cases, the distillation cost becomes comparable to the gate synthesis cost. Our proof of Thm. 2 reduces it to a matrix factorization problem, which can be solved using a known algorithm. This is remarkable because the optimisation problem for is believed to be a hard problem, see Ref. Amy and Mosca (2016) and Sec III.1. We prove Thm. 2 in Sec. III.2.
Since finding the optimal is difficult, we need efficient algorithms for nearoptimal decompositions. We will show that a fast algorithm exists giving approximation solutions
Theorem 3
Let acting on qubits. There exists a algorithm that finds a decomposition of in terms of Clifford+ gates, with uses of gates where
(9) 
Previous efficient algorithms do not have such scaling. For instance, Amy et al. (2014) has no proven upper bound in count, though in practice may perform well. Implicit in Ref. Amy and Mosca (2016) is an efficient algorithm with a maximum cost, but this still leaves a significant gap compared to the scaling of optimal solutions.
While it is believed that in general the optimal gate synthesis problem is hard, special cases can be tractable. In Sec. III.4 we consider controlledunitaries in and show this subclass can be solved efficiently and optimally, with upper bounded by for qubit unitaries.
We also observe that does not behave additively, so there are unitaries and such that . While it is clear that composed gates can be subadditive in cost, it seems remarkable that entirely disjoint circuits enjoy a reduction in resource costs. This subadditivity is reminiscent of similar phenomena seen in different resource theoretic settings.
In the final section we tackle concrete applications. Previous results show error Toffoli gates are possible using 8 states. We find error suppressed Toffoli gates are available at an asymptotic cost of 6 states each, which is partly due to aforementioned subadditivity. As a mainly pedagogical exercise we consider many controlS gates. Last we consider a family of circuits composed of CCZ gates, where optimal gatesynthesis offers a saving of naive gatesynthesis, and we obtain a further factor 3 reduction in resource by using synthillation.
Ii The general framework
ii.1 The Clifford hierarchy and Clifford equivalence
Here we review the Clifford hierarchy, introduce an equivalence relation and fix some notation. The level of the Clifford hierarchy is defined as
(10) 
where is the Pauli group and we terminate the recursion with . The familiar Clifford group is . For higher levels of the hierarchy we get nonClifford gates. Here we concern ourselves with nonCliffords from the third of the hierarchy. Specifically, we have defined the group , which is readily verified to be the diagonal subgroup of . Furthermore, we have that for all , the gate is in the diagonal Clifford group. In terms of weighted monomials we have . We give further details in App. A. The Clifford hierarchy is important as it has been shown that gates in can be performed by teleportation using Clifford operations and a particular resource state Gottesman99; Zhou et al. (2000). When the gate is also diagonal this resource is simply .
We say two unitaries and are Clifford equivalent whenever there exist Cliffords and such that . Since is a Clifford for any weighted polynomial , we know that and are Clifford equivalent. In other words, two unitaries and are Clifford equivalent whenever there exists an such that . In such cases we write where is an equivalence relation. It follows immediately that if then , and we reiterate that was specified in Def. 1. Since and are closely related, it is natural to ask whether (recall Def. 2) is related to some equivalence relation? In Sec. III.2 we introduce such an equivalence relation. Lastly, we use to denote the number of columns in matrix and to denote the number of rows in matrix .
ii.2 Quantum codes, encoders and quasitransversality
Central to synthillation are quantum codes with a special property we call quasitransversality. Here we define a quantum code in the matrix formalism, generalising the work of Bravyi and Haah Bravyi and Haah (2012). To specify a code we use a binary matrix partitioned into and .
Definition 3
Let be a binary matrix that is full rank with columns and rows that is partitioned into and so that . We define a quantum code with logical basis states
(11) 
This is an code where is the number of columns in , is the number of rows in , and with some distance .
We note that the element of is explicitly
(12) 
We say the code is trivial if the partition is empty, which entails . Bravyi and Haah considered binary matrices split according to row weight, with odd weight rows in and even weight rows in . We do not make this assumption, but will later impose a more complex condition dependent on the desired unitary.
Next, we review properties of encoder circuits used to prepare these quantum codes states. We use that for any invertible binary matrix , there exists Dehaene and De Moor (2003); Patel et al. (2003); Maslov (2007) a CNOT circuit such that
(13) 
In addition to its action on the computational basis, we track how these unitaries alter Pauli operators. To describe operators acting on many qubits we use where is some binary vector. Therefore,
(14) 
where throughout is the inner product satisfying . The Clifford affects the conjugation
(15)  
We use that the inner product satisfies to conclude that
(16) 
For a quantum code, the matrix will not be square, and so cannot be invertible. However, there will always exist an invertible that completes , so that
(17) 
for some . We consider to act on a partitioned bit string composed of , , and , so that
(18) 
and
(19) 
and similarly
(20) 
For the special case , we have and so
(21)  
Therefore, with appropriate ancilla qubits set to , all completions of behave identically, independent of the choice of . From here onwards, we use to denote any unitary with the above action. We will often refer to as an encoder for the quantum code associated with because of the following
(22) 
This shows how logical stabilizer states can be prepared using unencoded stabilizer states and CNOT gates.
Crucially important are quantum codes with the following property.
Definition 4
Let be a weighted polynomial and the associated unitary. We say a quantum code is quasitransversal if there exists a Clifford such that acting on the code realises a logical .
Transversal logical gates can be realised with product unitaries. Here only the nonClifford part is required to have product form, and the Clifford gate can be nonproduct, so we say they are quasitransversal. A sufficient condition for quasitransversality is the following.
Lemma 1
Let be a weighted polynomial with associated . Let be a by full rank matrix partitioned into and . The associated quantum code is quasitransveral if
(23) 
Here we use to denote the weight of a vector, so . Before proving the lemma, let us unpack the notation. The equation is evaluated , but is always evaluated . Furthermore, this compact notation can be expanded out as
(24) 
Applying to an encoded state gives
(25) 
Any diagonal Clifford acts as
(26) 
for some , where we set some qubits zero. We define another diagonal Clifford so that
(27)  
Therefore, the combined unitary acts as
(28) 
The lemma assumes that , which is equivalent to the existence of an such that
(29) 
Furthermore, since , the exponent of is can be taken modulo 8, and so
(30) 
Using this to specify and thereby , we have
(31) 
Since the phase no longer depends on , the phase can come outside the summation
(32) 
This proves quasitransversality follows from the condition stated in the lemma.
ii.3 The synthillation protocol
Given a quasitransversal quantum code, we can construct protocols for preparing magic states.
Theorem 4
Let be a by full rank matrix. Let so that the associated quantum code is quasitransversal. There exists a distillation protocol using only Clifford operations and noisy states with error rate . The protocol outputs the magic state with error rate where is the distance of the quantum code associated with . If , then the success probability is .
The above is a key finding of this work, and essentially delegates the task of finding synthillation protocols to finding matrices with the required properties. One can express BravyiHaah’s notion of triorthogonality as
(33) 
and so our concept of quasitransversality is a generalisation thereof. We discuss this point further in App. D.
We describe the protocol as a quantum circuit in Fig. 3. We first show why the protocol works in the absence of noise. Up to step 5 we have,
(34)  
which follows directly from quasitransversality. Without noise, the measured qubits are in the state and so yield “+1” outcomes in step 6. After discarding qubits in step 7 we are left with .
Now we consider noise. The noisy gates can, by twirling, be ensured to only suffer from Pauli noise. Therefore, at step 3 we must add the operator with probability . Recalling Eq. (20) and using that commutes with we have
(35)  
where . Therefore, the noisy output differs by from the ideal case (see Eq. 34) so that
(36)  
where between the second and last line we have used to eliminate . Some Pauli operators will act nontrivially on the qubits, flagging up the error. In step 6, we measure the qubits in the state obtaining the SUCCESS outcome only if and so . Therefore, the success probability is
(37) 
The output state is which is the correct state whenever and so . Therefore, the normalised error rate is
(38) 
For a distance code, we have that if and then . This allows us to conclude the scaling , and completes our proof of Thm. (4).
Given a matrix , the above expressions allow us to find the exact expressions for and by summing over all meeting the criteria. Typically, this sum will involve many terms and for large matrices could be computationally challenging. However, the sums can be reduced to far fewer terms by using the MacWilliams identities to move to a dual picture. This use of MacWilliams identities is a standard trick used within the field Bravyi and Kitaev (2005); Bravyi and Haah (2012); Campbell et al. (2012) and entails
(39) 
where we sum over all bits strings in the vector space generated by the rows of , which we denote as .
ii.4 Constructing the codes
Thm. 4 showed how to perform synthillation given matrices satisfying certain conditions depending on the target unitary . The next step in our proof is to construct such matrices from submatrices. We begin by introducing the building blocks.
Definition 5
We say a binary matrix is a gatesynthesis matrix for unitary if
(40) 
where is the weighted polynomial for .
This definition is a simpler version of the quasitransversality condition of Eq. (23), because a gatesynthesis matrix does not have the additional degrees of freedom needed to suppress errors. Throughout, we use to denote a gate synthesis matrix for , so . Recall that Thm. 4 and Def. 2 made use of a decomposition where , and so we use to denote the gate synthesis matrix for any such , so that . We find later that optimal matrices have columns numbering and . The next section discusses techniques for constructing and , and to what extent optimal constructions can be found by an efficient algorithm. However, for the purposes of this section, these matrices need not be optimal. If suboptimal matrices are used, the resource cost is .
The construction of distillation matrix and the value of the constant vary depending on numerous features, leading to 11 different cases presented in Table 1. Here we give an explicit proof of the result for three cases of increasing complexity. The remaining cases follow the same methodology with only minor changes. Before we begin the proofs, we review some of the basic tools. For any weighted polynomial of the form , we have that

;

and so for homogeneous cubic functions ;

and so for functions without a linear component ;

if and , then ;
Property 1 follows directly from the discussion in Sec. II.1. Property 2 and 3 follows directly due to modulo 8 arithmetic. The last property is also proven by similar expansions and degree counting. We shall also make use of the modular identity where is the elementwise product of two vectors.
We begin by considering the simple case 9
where throughout and the vector length should be clear from the context. We remind the reader that bold font symbols are used for column vectors, and so row vectors carry a transpose. We have
Using the modular identity, we have
(41) 
We notice that , where . We assume for case 9 that , which ensures
(42) 
Next, we rearrange the last term
and use for all , so that
The matrix is assumed to satisfy , and using this we have
Since and , we have by property (4) that , and so
(43) 
Combining the above equations gives
(44) 
The above expressions hold for all unitaries and will be reused later. We now consider the special case where is homogeneous cubic, and so by property (2) we have . This entails
(45) 
which is the desired result.
Next, we tackle the more general case where is not a CCZ circuit, but the weighted polynomial still has no linear terms. Let us consider case 5, and so assume and , and set
The weight now has three main contributions
(46)  
We can reuse Eq. (44), and make similar derivations for the terms, so that
(47)  
The function appears twice, but . Using , we deduce that
where we have used property (4) in moving to the second line. Combining these observations and regrouping terms gives
We know that only differs from by cubic terms, and so for some linear, quadratic and cubic polynomials. Since , we know by property (4) that . Although is not Clifford, it is homogeneous cubic and so by property (2) vanishes when multiplied by . Therefore, and so
(48) 
Applying property (3) we have
(49) 
The above has not yet assumed any special properties of the unitary and will be reused later. Now we use that has no linear terms, so . This completes the proof of quasitransversality for case 5.
CASE 1  CASE 5  
is even, is even  
CASE 2  CASE 6  
is even, is odd 