aa r X i v : . [ qu a n t - ph ] S e p Circuit Relations for Real Stabilizers:Towards
TOF + H Comfort, ColeOctober 2, 2019
Abstract
The real stabilizer fragment of quantum mechanics was shown to have a complete axiomatizationin terms of the angle-free fragment of the ZX-calculus. This fragment of the ZX-calculus—althoughabstractly elegant—is stated in terms of identities, such as spider fusion which generally do not haveinterpretations as circuit transformations.We complete the category
CNOT generated by the controlled not gate and the computational an-cillary bits, presented by circuit relations, to the real stabilizer fragment of quantum mechanics. Thisis performed first, by adding the Hadamard gate and the scalar √ The angle-free fragment of the ZX calculus—describing the interaction of the Z and X observables, Hadamardgate and π phases—is known to be complete for (pure) real stabilizer circuits (stabilizer circuits with realcoefficients) [15]. In [15, Section 4.1], it is shown that real stabilizer circuits are generated by the controlled- Z gate, the Z gate, the Hadamard gate, | i state preparations and h | post-selected measurements. Therefore,real stabilizer circuits can also be generated by the controlled-not gate, Hadamard gate, | i state preparationsand h | post-selected measurements. Although the Hadamard gate, controlled-not gate and computationalancillary bits are derivable in the angle-free fragment of the ZX-calculus, the identities are not given interms of circuit relations involving these gates: and instead, on identities such as spider fusion which do notpreserve the causal structure of being a circuit. Therefore circuit simplification usually involves a circuitreconstruction step at the end.We provide a complete set of circuit identities for the category generated by the controlled-not gate,Hadamard gate and state preparation for | i and postselected measurement for h | and the scalar √ T counts [21]. This optimization is performed in three steps: first, a translation must be performed turningthe circuit into spiders and phases. Second, the spider fusion laws, Hopf laws, bialgebra laws and so on are1pplied to reduce the number of nodes/phases; transforming the circuit into a simpler form resembling anundirected, labeled graph without a global causal structure. Finally, an optimized circuit is re-extractedfrom this undirected graph. In order to extract circuits at the end, for example, [13, 14] use a property ofgraphs called gFlow.Using only circuit relations, in contrast, [16] were able to reduce 2-qubit Clifford circuits to minimalforms in quantomatic.Toffoli+Hadamard quantum circuits, as opposed to the ZX-calculus, are more suitably a language for clas-sical oracles, and thus, are appropriate for coarse granularity optimization. The controlled-not+Hadamardsubfragment, on the other hand, which we discuss in this paper one can only produce oracles for affineBoolean functions—which is obviously very computationally weak. The eventual goal, however, is to usethis complete axiomatization controlled-not+Hadamard circuits given in this paper, and the axiomatizationof Toffoli circuits provided in [9], to provide a complete set of identities for the approximately universal [2]fragment Toffoli+Hadamard circuits. In this fragment, indeed, all oracles for classical Boolean functions canbe constructed [9, 1]. In Section 6, we discuss how this circuit axiomatization of controlled-not+Hadamardcircuits could potentially lead to one for Toffoli+Hadamard circuits,Toffoli+Hadamard circuits also easily accommodate the notion of quantum control. This is useful forimplementing circuits corresponding to the conditional execution of various subroutines; which is discussedin [17, Section 2.4.3] and [19]. Although, in the fragment which we discuss in this paper, we can not controlall unitaries: namely circuits containing controlled-not gates can not be controlled. Again, the eventual goalis to extend the axiomatizations of cnot+Hadamard and Toffoli circuits to Toffoli+Hadamard circuits, wherethere is no such limitation.In the ZX-calculus, by contrast, this notion of control is highly unnatural. One would likely have toappeal to the triangle gate, as discussed in [23, 20, 25]. Recall that
CNOT is the PROP generated by the 1 ancillary bits | i and h | as well as the controlled notgate: | i := | i := cnot :=Where “gaps” are drawn between cnot gates and cnot gates are drawn upside down to suppress symmetrymaps: :=These gates must satisfy the identities given in Figure 1:2 CNOT.1] = [CNOT.2] = [CNOT.3] = [CNOT.4] == [CNOT.5] = [CNOT.6] = [CNOT.7] == [CNOT.8] = [CNOT.9] =Figure 1: The identities of CNOT
Where the not gate and | i ancillary bits are derived: not = := | i = := h | = :=Where there is a pseduo-Frobenius structure (non-unital classical structure) generated by::= and :=There is the following completeness result: Theorem 2.1.
CNOT is discrete-inverse equivalent to the category of affine partial isomorphisms betweenfinite-dimensional Z vector spaces, and thus, is complete. In this section, we briefly describe the well known fragment of quantum mechanics known as stabilizerquantum mechanics. In particular we focus on the real fragment of stabilizer mechanics, and describe acomplete axiomization thereof called the angle-free ZX-calculus. Stabilizer quantum mechanics are very wellstudied, a good reference from a categorical perspective is given in [5].
Definition 3.1.
The
Pauli matrices are the complex matrices: X := (cid:20) (cid:21) Y := (cid:20) − ii (cid:21) Z := (cid:20) − (cid:21) The
Pauli group on n is the closure of the set: P n := { λa ⊗ · · · ⊗ a n | λ ∈ {± ± i } , a i ∈ { I , X , Y , Z }} under matrix multiplication.The stabilizer group of | ϕ i denoted by S | ϕ i a quantum state is the group of operators for which | ϕ i isa +1 eigenvector. A state is a stabilizer state in case it is stabilized by a subgroup of P n . Definition 3.2.
The
Clifford group on n is the group of operators which acts on the Pauli group on n byconjugation: C n := { U ∈ U (2 n ) |∀ p ∈ P n , U pU − ∈ P n } Lemma 3.3.
All n qubit stabilizer states have the form C | i ⊗ n , for some member C of the Clifford groupon n qubits.Indeed, we also consider a subgroup of C n : Definition 3.4.
The real Clifford group on n qubits, is the subgroup of the Clifford group with realelements, ie: C ren := { U ∈ C n | U = U } So that an n -qubit real stabilizer state is a state of the form C | i ⊗ n for some real Clifford operator C . We say that a (real) stabilizer circuit is a (real) Clifford composed with state preparations andmeasurements in the computational basis.The ZX-calculus is a collection of calculi describing the interaction of the complementary Frobeniusalgebras corresponding to the Pauli Z and X observables and their phases. The first iteration of the ZX-calculus was described in [10]. Definition 3.5. A Frobenius algebra in a monoidal category is a 5-tuple:( A , , , , )such that ( A , , ) is a monoid and ( A , , ) is a comonoid:= = = == =And the Frobenius law holds: [F] = =A Frobenius algebra is special if =and commutative if the underlying monoid and comonoids are commutative and cocommutative:= =A † -Frobenius algebra ( A , , ) is a Frobenius algebra of the form( A , , , † , † )That is to say, the monoid and comonoid are daggers of each other.Special commutative † -Frobenius algebras are called classical structures .A non-(co)unital special commutative † -Frobenius algebra is called a semi-Frobenius algebra . Semi-Frobenius algebras are used to construct a weak product structure for inverse categories such as CNOT and
TOF .However, we are interested in a simple fragment of the ZX-calculus, namely the angle-free calculus forreal stabilizer circuits, ZX π , described in [15] (slightly modified to account for scalars):4 efinition 3.6. Let ZX π denote the † -compact closed PROP with generators:such that ( , , , )is a classical structure, corresponding to the Z basis, and the following identities also hold up to swappingcolours: [PP] π π = [PI] π = πππ = ππ [B.U’] == [B.H’] = [B.M’] = [H2] = [ZO] π = π [IV] = [L] = == = [S1] π := [S2] ... α ... := ... α ... [S3] ... ... := ...... Figure 2: The identities of ZX π (where α ∈ { π } )The last 3 Axioms are actually definitions, which simplify the presentation of ZX π . Note that the axiomsof a classical structure are omitted from this box to save space.These axioms imply that the black and white Frobenius algebras are complementary where the antipodeis the identity.This category has a canonical † -functor, as all of the stated axioms are horizontally symmetric. It is also † -compact closed.This category embeds FHilb ; the black Frobenius algebra corresponds to the Pauli Z basis; the whiteFrobenius algebra corresponds to the Pauli X basis; the gate corresponds to the Hadamard gate and π and π correspond to Z and X π -phase-shifts respectively. In particular, the
X π -phase-shiftis the not gate.Because the π -phases are given by [S3] , by the commutative spider theorem, it is immediate that theyare phase shifts: 5 PH] π = ππ = π In bra-ket notation, a black spider from n to m with angle θ is interpreted as follows in FHilb : | i ⊗ n h | ⊗ m + e iθ | i ⊗ n h | ⊗ m and a white spider from n to m with angle θ is interpreted as follows in FHilb : | + i ⊗ n h + | ⊗ m + e iθ |−i ⊗ n h−| ⊗ m Note that the controlled-not gate has a succinct representation in ZX π (this can be verified by calculation):This means that ZX π contains all of the generators of the real Clifford group. Furthermore, the followingis known: Theorem 3.7. [15] ZX π is complete for real stabilizer states.The original presentation of ZX π in [15] did not account for scalars; instead, it imposed the equivalencerelation on circuits up to an invertible scalar and ignored the zero scalar entirely. Therefore, the originalcompleteness result described in [15] is not actually as strong as Theorem 3.7. This means, of course, thatthis original calculus does not embed in Mat C as the relations are not sound. For example, the followingmap is interpreted as √
2, not 1, in
Mat C :Later on, [4, 7] showed that by scaling certain axioms to make them sound, and by adding Axioms [IV] and [ZO] this fragment of the ZX-calculus is also complete for scalars. The properly scaled axioms have allbeen collected in Figure 2. CNOT into ZX π Consider the interpretation of
CNOT into ZX π , sending:
7→ 7→ π π We explicitly prove that this interpretation is functorial.
Lemma 4.1.
The interpretation of
CNOT into ZX π is functorial. Proof.
See A 4.1Because the standard interpretations of
CNOT and ZX π into Mat C commute and are faithful, the followingdiagram of strict † -symmetric monoidal functors makes CNOT → ZX π faithful: CNOT $ $ $ $ ■■■■■■■■ (cid:15) (cid:15) ZX π / / / / Mat C Extending
CNOT to ZX π As opposed to the ZX-calculus, the identities of
CNOT are given in terms of circuit relations . When applyingrules of the ZX calculus, circuits can be transformed into intermediary representations so that the flow ofinformation is lost. Various authors have found complete circuit relations for various fragments of quantumcomputing. Notably, Selinger found a complete set of identities for Clifford circuits (stabilizer circuits withoutancillary bits) [24]. Similarly, Amy et al. found a complete set of identities for cnot-dihedral circuits (withoutancillary bits) [3].In this section, we provide a complete set of circuit relations for real stabilizer circuits (although circuitscan have norms greater than 1). We show that CNOT is embedded in ZX π and we complete CNOT to ZX π by adding the Hadamard gate and the scalar √ Definition 5.1.
Let
CNOT + H denote the PROP freely generated by the axioms of CNOT with additionalgenerators the Hadamard gate and √ √ satisfying the following identities: [H.I] = [H.F] = [H.L] = [H.Z] √ = [H.S] √ =Figure 3: The identities of CNOT + H (in addition to the identities of CNOT )The inverse of √ / √ := [H.I] is stating that the Hadamard gate is self-inverse. [H.F] reflects the fact that composing thecontrolled-not gate with Hadamards reverses the control and operating bits. [H.Z] is stating that √ [H.S] makes √ / √ [H.L] can be restated to resemble [S1] : Lemma 5.2.[H.L’] = Proof. = ∆ is commutative== [H.F] , [H.I] = [H.L’] H.Z] can be restated in slightly different terms:
Lemma 5.3.[H.Z’] = Proof.
Immediate by [H.S] and [H.Z] .There is a derived identity, showing that the Frobenius structure identified with the inverse products of
CNOT is unital:
Lemma 5.4.[H.U] √ = = √ Proof. √ = √ = √ ∆ is commutative= √ [ H . I ], [ H . F ]= √ [ CNOT .7]= √ [ H . I ]= [ H . S ] CNOT + H We construct two functors between
CNOT + H and ZX π and show that they are pairwise inverses. Definition 5.5.
Let F : CNOT + H → ZX π , be the extension of the interpretation CNOT → ZX π whichtakes: √
7→ 7→
Let G : ZX π → CNOT + H be the interpretation sending:
7→ 7→ √
7→ 7→ √ The scalars √ n are taken to their chosen representatives by F and G : Lemma 5.6. (i) √ (ii) Proof.
See Lemma A.1This lemma makes it easier to show that F and G are functors: Lemma 5.7. F : CNOT + H → ZX π is a strict † -symmetric monoidal functor. Proof.
See A.1 Lemma A.2For the other way around:
Lemma 5.8. G : ZX π → CNOT + H is a strict † -symmetric monoidal functor. Proof.
See Lemma A.3Next:
Proposition 5.9.
CNOT + H F −→ ZX π and ZX π G −→ CNOT + H are inverses. Proof.
See Proposition A.4Because all of the axioms of
CNOT + H and ZX π satisfy the same “horizontal symmetry”; we can notonly conclude that they are isomorphic, but rather: Theorem 5.10.
CNOT + H and ZX π are strictly † -symmetric monoidally isomorphic. Recall the PROP
TOF , generated by the 1 ancillary bits | i and h | (depicted graphically as in CNOT ) aswell as the Toffoli gate: tof :=The axioms are given Figure 4, which we have put in the Appendix B. Recall that have that:
Theorem 6.1.
TOF is discrete-inverse equivalent to the category of partial isomorphisms between finitepowers of the two element set, and thus, is complete.By [1], we have that the Toffoli gate is universal for classical reversible computing, therefore
TOF is acomplete set of identities for the universal fragment of classical computing. However, the category is clearlyis not universal for quantum computing. Surprisingly, by adding the Hadamard gate as a generator, thisyields a category which is universal for an approximately universal fragment of quantum computing [2].Thus, one would hope that the completeness of
CNOT + H could be used to give a complete set ofidentities for a category TOF + H .Although we have not found such a complete set of identities, the identity [H.F] can be easily extendedto an identity that characterizes the commutativity of a multiply controlled- Z -gate. This could possiblyfacilitate a two way translation to and from the ZH calculus [6], like we performed between CNOT + H and ZX π . This, foreseeably would be much easier than a translation between one of the universal fragmentsof the ZX-calculus; because, despite the recent simplifications of the Toffoli gate in terms of the triangle,the triangle itself does not have a simple representation in terms of the Toffoli gate, Hadamard gate andcomputational ancillary bits [25].If we conjugate the not gate ( X gate) with Hadamard gates, we get the Z gate::=Furthermore, if we conjugate the operating bit controlled-not gate with the the Hadamard gate, we getthe controlled- Z gate: := 9ecause the flow of information in the controlled- Z gate is undirected in the sense that:=this motivates the identity [ H . F ] of CNOT + H := = = = =We can continue this, so that by conjugating the operating bit of the Toffoli gate with Hadamard gates,we obtain a doubly controlled- Z gate. This suggests the following (sound) identity: [H.F’] =Along with [TOF.16] , this entails: =So that we can unambiguously represent the doubly controlled- Z gate as:Indeed, this identity entails a more general form for generalized controlled-not gate with two or morecontrols. Recall the definition of a multiply controlled-not gate in [9]: Definition 6.2. [9, Definition 5.1] For every n ∈ N , inductively define the controlled not gate, cnot n : n + 1 → n + 1 inductively by: • For the base cases, let cnot := not , cnot := cnot and cnot := tof . • For all n ∈ N such that n ≥ cnot n +1 ≡ n :=Recall cnot n gates can be decomposed into other cnot n gates in the following fashion: Proposition 6.3. [9, Proposition 5.3 (i)] cnot n + k gates can be zipped and unzipped: nk = nk Therefore, can can derive that:
Lemma 6.4. [H.F’] entails: =
Proof.
From the zipper lemma and [H.F’] , we have:= = =10ecall from [9, Corollary 5.4] that control-wires of cnot n gates can be permuted in the following sense:=Therefore, by this observation and Lemma 6.4 we have:=So that, by observing that the multiply controlled- Z gate can be unambiguously defined as follows::= Acknowledgement
The author would like to thank Robin Cockett, Jean-Simon Lemay and John van de Wetering for usefuldiscussions.
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CNOT into ZX π (proofs) The following basic identity is needed:
Lemma A.1. [25, Lemma 19] =
Proof. === [ L ] × L ]= [ B . U ′ ]=We also need: Lemma A.2. (i) π (ii)
7→ 7→
Proof. (i) ππ = π [ IV ]= π [ B . U ′ ]= π = π Lemma A.1= π [ IV ](ii) := 13 → ππ π = π π [ PI ]= π π [ PH ]= π π [ B . U ′ ]= [ PP ]== Lemma A.1, [ IV ]= Lemma 4.1.
The interpretation of
CNOT into ZX π is functorial. Proof.
We prove that each axiom holds [CNOT.2] = (co)associativity= [ B . H ′ ]14 (co)unitality ← [ [CNOT.1] == [ B . M ′ ]= ← [ [CNOT.3] Immediate from commutative spider theorem. [CNOT.4] Lemma A.2 (ii)= [ B . U ′ ]= ← [ Lemma A.2 (ii) [CNOT.5]
Immediate from commutative spider theorem. [CNOT.6]
Frobenius algebra is special. [CNOT.7] π = ππ [ PI ], [ B . U ′ ]15 ππ [ PH ] ← [ Lemma A.2 (i) [CNOT.8] == [ B . M ′ ]==== [ B . H ′ ] ← [ [CNOT.9] π Lemma A.2 (ii)= π [ ZO ]16 π Lemma A.1= π = π [ B . U ′ ]= ππ By symmetry= π ππ = π ππ [ ZO ] ← [ Lemma A.2 (ii)
A.1 Proof of Lemma 5.6
Lemma 5.6 . (i) √ (ii) (iii) Proof. (i) √ √ = √ [ H . S ](ii) √ √ = √ √ √ √ [ H . F ]= √ √ [ CNOT .1], [
CNOT .2]= √ √ = √ √ [ H . F ]= √ √ = √ √ [11, Lemma B.0.2]= √ √ / √ / √ / √ = / √ [ H . S ]=(iii) = [ H . S ] A.2 Proof of Lemma 5.7
We show that these interpretations are functors:
Lemma 5.7. F : CNOT + H → ZX π is a strict † -symmetric monoidal functor. Proof.
The preservation of the † -symmetric monoidal structure is immediate. As the restriction of F to CNOT is a functor, it suffices to show that [H.I] , [H.F] , [H.U] , [H.L’] , [H.S] and [H.Z’] hold. [H.I] Immediate. [H.F] ← [ [H.L’] Immediate. [H.S]
Immediate. [H.Z’] π Lemma A.2 (ii)= π = π [ ZO ]= π Lemma A.1= π [IV] = π [ZO] = π [ IV ]= ππ π ππ [ PP ]= ππ ππ π [ ZO ]19 ππ πππ = ππ πππ [ B . U ′ ]= ππ ππ π [ PH ]= ππ π [ PP ]= ππ π [ ZO ]= ππ π ← [ Lemma A.2 (ii)
A.3 Proof of Lemma 5.8
Lemma 5.8 . G : ZX π → CNOT + H is a strict † -symmetric monoidal functor. Proof.
We prove that each axiom holds: [PI]
This follows by naturality of ∆ in
CNOT . [B.U’] This follows by naturality of ∆ in
CNOT and [H.S] . [H2] This follows immediately from [H.I] . [H2] This follows immediately from [H.S] . [PP] π π
20 [ H . I ]== [ H . I ] ← [ [B.H’] = [ H . F ]= [ CNOT .2]= √ √ / √ / √ [ H . Z ′ ] ← [ Lemma 5.6 (ii) × [B.M’] == [ H . F ]= ∆ natural in CNOT = / √ [ H . U ]= / √ [ H . F ]= / √ ← [ Lemma 5.6 (ii) [L] H . F ] ← [ [ZO] π √ √ = √ √ [ H . I ]= √ √ / √ = √ [ H . S ]= √ [ CNOT .9]= √ [ CNOT .7]= √ = √ √ [ H . L ′ ]= √ √ [ CNOT .9] × √ √ = √ √ [ CNOT .7] × √ √ / √ / √ [ CNOT .3] ×
4= [ H . S ] × √ √ √ As before22 √ √ √ √ / √ [ H . S ]= √ √ √ √ [ H . L ′ ] ← [ π Classical structure:
Remark that rules [H.U] and [H.S] complete the semi-Frobenius structure to theappropriate classical structure.
A.4 Proof of Proposition 5.9
Proposition 5.9.
CNOT + H F −→ ZX π and ZX π G −→ CNOT + H are inverses. Proof.
1. First, we show that G ; F = 1 .We only prove the cases for the generators cnot and | i as the claim follows trivially for the Hadamardgate and by symmetry for h | : For | i : π √ / √ Lemma 5.6 (ii)= [ H . S ] For cnot : F = G √ Lemma 5.6 (i)= √ [ H . F ]23 √ [ CNOT .9]= √ [ H . F ]= [ H . U ]= [ H . I ]= [ CNOT .2]= [
CNOT .1]=It is trivial to observe, on the other hand, that G ; F = 1. B The identities of
TOF
TOF is the PROP generated by the 1 ancillary bits | i and h | as well as the Toffoli gate: | i := | i := tof :=Where there is the derived generator cnot = :=and the not gate and | i ancillary bits are dervied as in Section 2. These generators must satisfy the identitiesgiven in the following figure: 24 TOF.1] = , = [TOF.2] = , = [TOF.3] = [TOF.4] = [TOF.5] = [TOF.6] = [TOF.7] = [TOF.8] = [TOF.9] = [TOF.10] = [TOF.11] = [TOF.12] = [TOF.13] = [TOF.14] = [TOF.15] = [TOF.16] = [TOF.17] =Figure 4: The identities of TOF
Errata
The author would like to apologize for giving an incorrect proof of Corollary 4.9 in the previous preprintversion; meaning that the scalar √√