pre-cisely to the edges of Kk? G and that we have
I2r(HstabG,k) = I2r(HKk?G∪ CKk?G).
Based on this, one can show the analogue of Lemma 8.13: If L is feasible for the program ξstabr (Kk ? G), then we have L(P
i,cxic) = k if and only if L = 0 on I2r(CKk?G). This lemma can be used to show the following result, whose proof is analogous to that of Proposition 8.14 and thus omitted.
Proposition 8.15. For every graph G and r ∈ N ∪ {∞} we have γrstab(G) = max{k : ξrstab(Kk? G) = k}.
We do not know whether the results of Propositions 8.14 and 8.15 hold for r = ∗, because we do not know whether the supremum is attained in the program defining the parameter ξ∗stab(·) = αp(·) (as was already observed in [Rob13, p. 120]). Hence we can only claim the inequalities
γ∗col(G) ≥ min{k : ξ∗stab(GKk) = |V |} and γ∗stab(G) ≤ max{k : ξ∗stab(Kk?G) = k}.
As mentioned above, we have lascolr (G) ≤ Λr(G) for every integer r ∈ N [GL08b, Prop. 3.3]. This result extends to the noncommutative setting and the analogous result holds for the stability parameters. In other words the hierarchies {γrcol(G)}
and {γrstab(G)} refine the hierarchies {ξcolr (G)} and {ξstabr (G)}.
Proposition 8.16. For every graph G and r ∈ N ∪ {∞, ∗} we have ξcolr (G) ≤ γrcol(G) and ξrstab(G) ≥ γrstab(G).
Proof. We may restrict to r ∈ N since we have seen earlier that the inequalities hold for r ∈ {∞, ∗}. The proof for the coloring parameters is similar to the proof of [GL08b, Prop. 3.3] in the classical case and thus we omit it. We now show ξstabr (G) ≥ γrstab(G). Set k = γrstab(G) and, using Proposition 8.15, let L ∈ Rhxic : i ∈ V, c ∈ [k]i∗2r be optimal for ξstabr (Kk? G) = k. That is, L is tracial, symmetric, positive, and satisfies L(1) = 1, L(P
i,cxic) = k, and L = 0 on I(HKk?G). It suffices now to construct a tracial symmetric positive linear form ˆL ∈ Rhxi: i ∈ V i∗2rsuch that ˆL(1) = 1, ˆL(P
i∈V xi) = k, and ˆL = 0 on I2r(HG), since this will imply ξstabr (G) ≥ k. For this, for any word xi1· · · xit with degree 1 ≤ t ≤ 2r, we define L(xˆ i1· · · xit) :=P
c∈[k]L(xic1· · · xict), and we set ˆL(1) = L(1) = 1. Then, we have L(ˆ P
i∈V xi) = k. Moreover, one can easily check that ˆL is indeed tracial, symmetric, positive, and vanishes on I2r(HG).
8.4 Discussion
Let us discuss some known separations between the quantum graph parameters and their classical analogues.
152 Chapter 8. Quantum graph parameters The separations between χq(G) and χ(G), and between αq(G) and α(G), can be exponentially large in the number of vertices. This is the case for the graphs with vertex set {±1}N for N a multiple of 4, where two vertices are adjacent if they are orthogonal [AHKS06, MR16b, MSS13]. These graphs are often called Hadamard graphs (notice that a clique of size N corresponds to a real Hadamard matrix, see Section 2.2).
Let us explain the separation between χq(G) and χ(G) for these graphs. Let N ∈ N and let GN = (V, E) be the graph where V = {±1}N and quantum coloring game. We use the state ψ = √1
N
PN
i=1ei⊗ ei ∈ CN ⊗ CN. To describe Alice’s and Bob’s POVMs it will be usefull to consider the unitary matrix ΩN = √1
N(ωijN)i,j∈[N ], where ωN = e2πi/N is the N th root of unity. This matrix is known as the discrete Fourier transform. For each question x ∈ V , Alice has a POVM {Aix}i∈[N ], and for each question y ∈ V , Bob has a POVM {Biy}i∈[N ], where
The claim is that the bipartite quantum correlation P corresponding to this state and these POVMs is a perfect strategy for the coloring game. To see this, we compute the probability that Alice and Bob output i and j respectively, when they are given questions x, y ∈ V :
In particular, the probability that Alice and Bob both answer i ∈ [N ] is
P (i, i|x, y) = 1
Therefore, if Alice and Bob receive adjacent vertices x and y, that is, ifPN
k=1xkyk = 0, then the probability that they both answer color i equals zero. Similarly, if Alice
2The same argument was given earlier for the case where N is a power of 2. See [BCT99] for the same argument in the setting of graph coloring, and see [BCW98] for a similar argument in a different setting.
8.4. Discussion 153 and Bob receive the same vertex x = y, then the above shows that P (i, i|x, x) = N1 for each i ∈ [N ]. It follows thatPN
i=1P (i, i|x, x) = 1 and therefore P (i, j|x, x) = 0 if i 6= j, that is, P is synchronous. Together this shows that P is indeed a perfect strategy for χq(G) and therefore χq(G) ≤ N .
In fact, one can show that χq(GN) = N . For this, we can use the theta number of the complement G of G. Recall that ϑ(G) ≤ χq(G). It was shown in [MR16b, Prop. 4.2] that if N is divisible by 4, then ϑ(GN) = N .
Finally, it follows from a result of Frankl and R¨odl [FR87, Thm. 1.11] that for large enough N divisible by 4, the chromatic number of GN is exponential in N . Informally, the result of Frankl and R¨odl implies that there are no large independent sets in GN. Therefore, the chromatic number needs to be large. Together, this shows that the ratio between χ(G) and χq(G) can be exponential in the number of vertices.
What about α(G) and αq(G)? In [MR16b] it is shown that the graphs GN can also be used to show an exponential separation between αq(G) and α(G).
Can we also separate the quantum parameters from their commuting operator analogues? This is still an open question. While it was recently shown that the sets Cq,s(Γ) and Cqc,s(Γ) can be different [DPP19], it is still not known whether there is a separation between the parameters χq(G) and χqc(G), and between αq(G) and αqc(G).
Let us finish by noting a remarkable property of the quantum chromatic number.
It is easy to see that the chromatic number of a graph increases by 1 if we add a new vertex that is adjacent to all other vertices. Surprisingly, this is not true in general for the quantum chromatic number [MR16a].
Part II
Quantum algorithms &
optimization
155
Chapter 9
Quantum algorithms
In this background chapter we introduce the basic concepts of quantum algorithms and we give an overview of the main quantum algorithms that will be used as subroutines in the subsequent chapters. For more details see for example [NC00], or the lecture notes [Wat11, dW11].
9.1 The basics
In Chapter 3 we have seen that the state of a quantum-mechanical system can be described by a unit vector in a Hilbert space, and that the allowed operations are applications of unitary operators and measurements. A quantum algorithm consists of precisely those operations: we start with an initial state ψ, we apply a unitary U to ψ and then we perform some m-outcome projective measurement {E1, . . . , Em} to U ψ.1 Below we introduce some notation and concepts that allow us to talk about the complexity of a quantum algorithm.
Dirac notation. The fundamental building block of a classical Boolean circuit is a bit, which is either 0 or 1. The quantum analogue is the quantum bit, a qubit, a superposition over two basis states, that is, a unit vector ψ ∈ C2. It will be useful to think of the standard basis vectors of C2 as ‘0’ and ‘1’. To emphasize this, for quantum algorithms we will use the Dirac notation for the standard basis vectors of Cd: |0i, . . . , |d − 1i. The conjugate transpose of a vector |ψi ∈ Cd is denoted by hψ|. We will use the shorthand notation |ψi|φi for the state |ψi ⊗ |φi. When |ψi and |φi are standard basis vectors we sometimes even use the notation |ψ, φi for
|ψi|φi. From now on, unless explicitly stated otherwise, a vector |ψi is assumed to be a normalized state: a unit vector in some complex Hilbert space.
Quantum circuits. We can describe a classical computation by a sequence of wires, which carry bits, and logical gates which act on those bits. For example we
1Note that a product of unitary operators is again a unitary operator. Also, without loss of generality we may assume that all measurements are deferred until the end [NC00, Section 4.4].
157
158 Chapter 9. Quantum algorithms can apply the NOT gate to a bit b which transforms it into b ⊕ 1, or we can take the OR of two bits a and b which evaluates to 0 if a = b = 0 and to 1 otherwise. Similarly a quantum computation can be described by wires, which now carry qubits, and some elementary gates that act on them. Two important gates are the Hadamard gate H and the controlled-not gate CNOT. The Hadamard gate acts on a single qubit |bi (where b ∈ {0, 1}) as
(i.e., if the control-qubit |ai is |0i then CNOT acts as the identity on the second qubit, otherwise it performs the NOT gate on it). As matrices the Hadamard and CNOT gate can be represented as follows (in the standard basis):
H = 1
For example, the Hadamard gate can be used to create a uniform superposition over all 2n standard basis vectors of (C2)⊗n:
As another example, we can construct the EPR-pair that we have seen in Section 3.1 (an entangled state!) from 2 qubits initialized in |0i|0i by first applying a Hadamard gate on the first qubit and then a CNOT gate on the second qubit where the first qubit acts as the control-qubit:
Complexity. The AND and NOT gate mentioned above are universal for classical computation, meaning that any Boolean function f : {0, 1}n → {0, 1} can be com-puted using a circuit containing only AND and NOT gates. Is there also a small uni-versal gate set for quantum computation? One can show that the set of all 1-qubit gates (unitaries acting on a single qubit) together with the CNOT gate is universal:
any unitary matrix can be written as a product of such small unitaries [NC00, Sec-tion 4.5.2]. The set of all 1-qubit gates is unfortunately rather big, but it turns out that it can be very efficiently approximated using only the Hadamard gate and the 1-qubit phase gate Rπ/4 defined as Rπ/4|0i = |0i, Rπ/4|1i = eiπ/4|1i. Indeed, the
9.2. The fundamental building blocks 159