OOverlapping qubits
Rui Chao Ben W. Reichardt Chris Sutherland Thomas Vidick Abstract
An ideal system of n qubits has 2 n dimensions. This exponential grants power, but alsohinders characterizing the system’s state and dynamics. We study a new problem: the qubits ina physical system might not be independent. They can “overlap,” in the sense that an operationon one qubit slightly affects the others.We show that allowing for slight overlaps, n qubits can fit in just polynomially manydimensions. (Defined in a natural way, all pairwise overlaps can be ≤ (cid:15) in n O (1 /(cid:15) ) dimensions.)Thus, even before considering issues like noise, a real system of n qubits might inherently lackany potential for exponential power.On the other hand, we also provide an efficient test to certify exponential dimensionality.Unfortunately, the test is sensitive to noise. It is important to devise more robust tests on thearrangements of qubits in quantum devices. Quantum computers start with the qubit, a two-level quantum system. They achieve their powerby combining many qubits. A system of n independent qubits is associated to a 2 n -dimensionaltensor-product space, ( C ) ⊗ n , and quantum algorithms exploit this exponential dimensionality.However, with great power also comes great guile. In experiments, it is exceedingly difficult tocharacterize the states and dynamics of large quantum systems. An efficient test, running inpolynomial time, can only probe a limited portion of an exponentially complex system.Before getting to state or process tomography, however, there is the problem of characterizingthe system’s Hilbert space, and the arrangement of the qubits within it. In particular, what if thequbits are not in tensor product, but “overlap”, so an operation on one qubit can slightly affectthe others? Given a system that supposedly has n independent qubits, how can we efficiently testthat there really are 2 n dimensions? Unfortunately, we show that very small systems, with onlypolynomially many dimensions, can contain n qubits that are nearly pairwise independent, i.e., anoperation on qubit i can have only a small effect on qubit j for all i (cid:54) = j . In fact, there are particularstates in n -dimensional systems for which n qubits look to be exactly pairwise independent, intensor product. (We will give more technical statements of these results in a moment.)The issue of overlapping qubits is a new concern for the characterization of quantum devices. Acommon complaint about today’s quantum devices, especially those targeted at adiabatic quantumoptimization or quantum annealing, is that it is difficult even to verify their quantum-ness [AHSL15].High noise rates can decohere systems, making them classical. Our examples raise a different University of Southern California Department of Computing and Mathematical Sciences, California Institute of Technology a r X i v : . [ qu a n t - ph ] J a n roblem: a system might indeed be quantum mechanical and even look like it has many qubits, butstill quantum power is lacking because the system is low-dimensional.On the other hand, we show that low-dimensional systems cannot totally fool us. First, if allpairs among n qubits are sufficiently close to being independent, then in fact there are nearby qubitsthat are exactly independent (in tensor product); and hence the dimension must be at least 2 n .Second, we provide a test for independence, one that efficiently checks not just pairwise interactionsbut n -wise interactions, and thereby can verify that the system dimension is almost 2 n . The testonly involves measuring the qubits one at a time, so it is conceivably practical—except it is stillsensitive to noise. Overlapping qubits
The concept of overlapping, dependent qubits is not standard in quantum information theory. Ingeneral, multiple qubits are always assumed to be in tensor product; in common usage n qubitsdirectly means ( C ) ⊗ n . However, though it may be invisibly built into our notation and habits ofthought, this is in fact an independence assumption, which needs to be justified. Precisely, then,what is a qubit, and what does it mean for two qubits to overlap?1. What is a qubit? A qubit in a space H is a two-dimensional register in tensor product withthe rest of the space. That is, from an isomorphism between H and C ⊗ H (cid:48) , the C registerdefines a qubit. Since the basis for H (cid:48) does not matter, instead of specifying the isomorphismit is more convenient to work in the dual Heisenberg picture, in which a qubit is definedthrough the observables that act on it, an algebra generated by the four Pauli matrices. Infact, a pair of norm-one observables X and Z that anti-commute suffice to define a qubit; it isthen possible to choose a basis in which X = σ x ⊗ H (cid:48) and Z = σ z ⊗ H (cid:48) , where σ x and σ z are the standard Pauli operators (see Lemma 2.2).2. Two qubits are independent, or in tensor product, when all operators on the qubits commute.Thus n qubits, defined by anti-commuting X j , Z j for j = 1 , . . . , n , are pairwise independent if[ X i , X j ] = [ X i , Z j ] = [ Z i , Z j ] = 0 for all i (cid:54) = j . It follows that there is a change of basis underwhich H = ( C ) ⊗ n ⊗ H (cid:48) and X j = σ xj ⊗ H (cid:48) , Z j = σ zj ⊗ H (cid:48) (Theorem 2.3).When are two qubits “almost” independent? For qubits specified by reflections X , Z and X , Z , how close they are to lying in tensor product can be measured by the largest commutatornorm, max S,T ∈{ X,Z } (cid:107) [ S , T ] (cid:107) .Almost independence is a useful concept because in reality one can never probe for the existence of n independent qubits. The exact tensor-product structure of a Hilbert space cannot be experimentallytested. Due to inevitable measurement imprecision, one could at best hope to show approximaterelations, like (cid:107) [ S i , T j ] (cid:107) ≤ (cid:15) . This concept is also mathematically well-motivated. It amounts tostudying approximate representations of the n -qubit Pauli group. It can alternatively be tied toquestions on the stability of relations defining the Pauli algebra [Lor93].
Our results
We begin by asking: how many overlapping qubits can be packed into 2 n dimensions? We proveboth lower and upper bounds. Of course, only n independent qubits fit. We caution that there does not seem to be a standard definition for an approximate group representation in themathematical literature; see, e.g., [BF91, MR15] for work in this direction. a) (b) Figure 1: (a) A qubit is a two-dimensional system in tensor product with the rest of the space.Qubits “overlap” if the corresponding Pauli operators do not commute. When their Pauli operatorsdo commute, the qubits are in tensor product with each other (Theorem 2.3). (b) We ask howmany qubits can be packed into a 2 n dimensional space with small pairwise overlap. For a lowerbound, we give a randomized construction, based on the Johnson-Lindenstrauss Lemma and fermionalgebra (Theorem 3.1). For an upper bound, we separate qubits with small pairwise overlap, findingnearby qubits with zero overlap (Theorem 3.6).For the lower bound, we give a randomized construction, based on the Johnson-Lindenstrausslemma, for packing many nearly orthogonal unit vectors, and on the exterior algebra. We show thatexponential in n many qubits can be packed with pairwise overlaps (cid:107) [ S i , T j ] (cid:107) of order (cid:112) (log n ) /n . Ingeneral, for overlaps (cid:107) [ S i , T j ] (cid:107) ≤ (cid:15) , e O ( n(cid:15) ) qubits can be packed into 2 n dimensions; see Theorem 3.1.Parameterized differently, the construction places n (cid:15) -overlapping qubits in only n O (1 /(cid:15) ) dimensions.Note that this construction does not allow for compressing information. Even though expo-nentially many nearly independent qubits can be packed into ( C ) ⊗ n , this does not allow forreliably storing more than n bits, and thus does not violate Nayak’s private information retrievalbound [Nay99]. If one tried to store (cid:29) n bits into ( C ) ⊗ n by putting a bit into each of the embeddedqubits, one at a time, by the end the early bits would be unrecoverable because of accumulatederrors.For the upper bound, we show that even allowing pairwise overlaps (cid:107) [ S i , T j ] (cid:107) as large as c/n , fora certain constant c , there is still room only for n qubits in 2 n dimensions. The precise statement isin Theorem 3.6. The proof constructively extracts n independent qubits from n overlapping qubits.The key difficulty is to ensure that errors do not explode; naively separating, say, the second qubitfrom the first could double its overlap with each of the remaining qubits, yielding an unmanageableexponential blow-up in the total displacement needed to separate the qubits. See Figure 1.The construction in the upper bound loses a factor of n , and we give an example to show thatthis is necessary (Lemma 3.9). Yet there is still a gap between our lower and upper bounds. For therange of overlaps 1 /n (cid:46) (cid:15) (cid:46) (cid:112) (log n ) /n , we do not know whether strictly more than n qubits canbe packed into 2 n dimensions.Given access to an experimental system, it is difficult to imagine tests for determining (cid:107) [ S i , T j ] (cid:107) .The problem is that the quantum system can be in an unknown state | ψ (cid:105) , and we can only learnabout operators’ effects on | ψ (cid:105) . If S i and T j are far from commuting, but only on a portion of theHilbert space in which | ψ (cid:105) has no support, this is undetectable. In Section 4, we therefore considera state-dependent overlap measure. This is the same measure that is used in results on self-testingsuch as [MY98, MYS12], and it is the relevant measure for applications to device-independentcryptography [KTW14]. Note however that our setting differs from the usual one in self-testing, aswe do not assume any a priori bipartite structure on the Hilbert space; though our results do applyto bipartite entanglement testing [CRSV16]. 3e first give a practical protocol for testing if (cid:107) [ S i , T j ] | ψ (cid:105)(cid:107) ≈
0: measure S i , measure T j , thenmeasure S i again and check that it gives the same result. However, this test is not enough; we give aconstruction of a state and n qubit operators in < n dimensions, such that for i (cid:54) = j , [ S i , T j ] | ψ (cid:105) = 0exactly. Finally we give a more advanced test that efficiently checks not just pairwise commutationrelationships, like [ S i , T j ] | ψ (cid:105) ≈
0, but also higher-order relationships like S i T j U k | ψ (cid:105) ≈ U k T j S i | ψ (cid:105) .This test can verify that the system dimension is almost 2 n . As explained in the introduction, we take a basis-independent, operator-centric view of whatit means to have a qubit, or multiple independent qubits, in an a priori unstructured Hilbertspace H . The following definition formalizes these notions. Notation: Let [ n ] = { , , . . . , n } , and I = (cid:0) (cid:1) , σ x = (cid:0) (cid:1) , σ y = (cid:0) − ii (cid:1) and σ z = (cid:0) − (cid:1) be the Pauli matrices. The commutator is[ S, T ] = ST − T S , and the anticommutator is { S, T } = ST + T S . When we write, e.g., “ S j for S ∈ { X, Z } ” we mean the set { X j , Z j } , i.e., the letter S is meant to be directly replaced by X or Z . Definition 2.1. A qubit in a Hilbert space H is a pair of anti-commuting reflections ( X, Z ) on H .The overlap between two qubits ( X , Z ) and ( X , Z ) is given by max S,T ∈{ X,Z } (cid:107) [ S , T ] (cid:107) . Thequbits are in tensor product if they have overlap ; in this case we also say that the qubits are independent . The following simple lemma ties this definition to the more usual one of a qubit as defined by afactorization
H (cid:39) C ⊗ H (cid:48) . The lemma is a special case of Theorem 2.3 below. Lemma 2.2.
Let X and Z be reflections (Hermitian operators that square to the identity) on aseparable Hilbert space H such that X and Z anti-commute: { X, Z } = 0 . Then there exists aseparable space H (cid:48) such that H is isomorphic to C ⊗ H (cid:48) , and up to a unitary change of basis thereflections X, Z are the standard Pauli operators: X = σ x ⊗ H (cid:48) , Z = σ z ⊗ H (cid:48) . The following theorem justifies our definition of two qubits being in “tensor product” when theiroverlap is 0, or equivalently when the associated reflections pairwise commute.
Theorem 2.3.
Suppose that X , Z , . . . , X n , Z n are reflections on H such that for all j , { X j , Z j } = 0 and furthermore for all i (cid:54) = j and S, T ∈ {
X, Z } , S i and T j pairwise commute, [ S i , T j ] = 0 . Thenthere exists a separable space H (cid:48)(cid:48) such that H is isomorphic to ( C ) ⊗ n ⊗ H (cid:48)(cid:48) , and up to a unitarychange of basis the reflections X j , Z j are the standard Pauli operators on n qubits: X = σ x ⊗ I ⊗ ( n − ⊗ H (cid:48)(cid:48) Z = σ z ⊗ I ⊗ ( n − ⊗ H (cid:48)(cid:48) · · · X n = I ⊗ ( n − ⊗ σ x ⊗ H (cid:48)(cid:48) Z n = I ⊗ ( n − ⊗ σ z ⊗ H (cid:48)(cid:48) . Proof.
Let X = X , Z = Z . As Z = , Π ± = ( ± Z ) are projections, with Π + + Π − = ,Π + − Π − = Z and Π + Π − = Π − Π + = 0. Multiplying both sides of { X, Z } = 0 by Π ± yieldsΠ ± X Π ± = 0, i.e., X = Π + X Π − + Π − X Π + . Then X = implies that Π ± X Π ∓ X Π ± = Π ± ; andcomparing the ranks of both sides gives Rank(Π ∓ ) ≥ Rank(Π ± ), i.e., Rank(Π + ) = Rank(Π − ).Let | u ± (cid:105) , | u ± (cid:105) , . . . be an orthonormal basis for Range(Π ± ). Let S = (cid:80) j ( | u + j (cid:105)(cid:104) u − j | + | u − j (cid:105)(cid:104) u + j | ).Then S = S † , S = and S Π ± = Π ∓ S . Let U = Π + X Π − S + Π − . U is unitary: U U † = U † U = .4urthermore, U † ZU = Z , and U † XU = S . Relabeling the basis elements | , j (cid:105) = | u + j (cid:105) , | , j (cid:105) = | u − j (cid:105) ,we obtain U † ZU = σ z ⊗ and U † XU = σ x ⊗ , as desired.Now consider X . In the above basis, it can be expanded as I ⊗ A + (cid:80) β ∈{ x,y,z } σ β ⊗ B β , butthe commutation relationships [ X , X ] = [ X , Z ] = 0 imply that each B β = 0. Similarly, all thereflections Z , . . . , X n , Z n act trivially on the first C register. Inductively repeating the aboveargument for X and Z gives the theorem.Registers that are in tensor product are independent of each other, in the sense that for a quantumstate | ψ (cid:105) ∈ H (cid:48) ⊗ H (cid:48)(cid:48) , a quantum operation on H (cid:48) cannot affect the reduced density matrix Tr H (cid:48) | ψ (cid:105)(cid:104) ψ | in the other register. It should be noted, though, that a qubit can simultaneously have maximaloverlap with many other mutually independent qubits. For example, for n odd, X = ( σ x ) ⊗ n and Z = ( σ z ) ⊗ n are anti-commuting reflections, defining a qubit, such that for every j ∈ [ n ], (cid:107) [ X, σ zj ] (cid:107) = (cid:107) [ Z, σ xj ] (cid:107) = 2. (Similarly, in ( C ) ⊗ n , for a Haar random unitary U , (cid:107) [ U σ α U † , σ βj ] (cid:107) willbe concentrated around the maximal value of 2.) Thus the norm of the reflections’ commutator isnot a “monogamous” measure of qubit overlap. How many pairwise (cid:15) -overlapping qubits can be packed into 2 n dimensions? Formally, in 2 n dimensions, we wish to place 2 m reflections ( X , Z ) , . . . , ( X m , Z m ) such that each pair ( X j , Z j )defines a qubit, so that { X j , Z j } = 0, and operators with different indices nearly commute: (cid:107) [ S i , T j ] (cid:107) ≤ (cid:15) for i (cid:54) = j and S, T ∈ {
X, Z } . How large can m be?One’s intuition might be pulled in either of two directions. From the perspective of informationtheory, Nayak’s private information retrieval bound m ≤ n/ (1 − H ( p )) [Nay99] suggests that packing ω ( n ) qubits into 2 n dimensions is unlikely to be possible. However, a formal connection between ourproblem and private information retrieval is not obvious: the existence of m pairs of approximatelycommuting qubit operators does not imply that there exists a family of 2 m states that could beused to encode m bits with a good probability of recovery.From a geometric perspective the problem can be viewed as one of packing subspaces. Eachreflection R j is about a certain subspace, projected to by ( I + R j ). As explained in the previoussection, the anticommutation condition implies that X j and Z j correspond to subspaces with allprincipal angles π/
4, while the approximate commutation condition (cid:107) [ S i , T j ] (cid:107) ≤ (cid:15) translates intothe corresponding subspaces making principal angles close to 0 or π/
2. By analogy to the problemof packing nearly orthogonal unit vectors one might guess that as long as (cid:15) is not required to go to0 too fast with n , m can be exponential in n .The results in this section demonstrate that the geometric intuition is more accurate. Theorem 3.2shows that for sufficiently small (cid:15) (inverse linear in n ), no more than m ≤ n (cid:15) -overlapping qubitscan fit in 2 n dimensions. In contrast, Theorem 3.1 shows that as long as (cid:15) = Ω(1), m can beexponential in n ; more generally m = ω ( n ) for any (cid:15) = ω ( (cid:112) (log n ) /n ). For the range of overlaps1 /n (cid:46) (cid:15) (cid:46) (cid:112) (log n ) /n , we do not know whether strictly more than n qubits can be packed into 2 n dimensions. For vector packing upper bounds on m , see, e.g., [KL78], [Alo03, Lemma 9.1], [Tao13]. .1 Lower bound: packing exponentially many qubits in n dimensions We give a randomized construction that packs m = e Θ( n(cid:15) ) qubits into 2 n dimensions. This beatsthe trivial m = n for (cid:15) = Ω( (cid:112) (log n ) /n ), and is exponential in n for constant (cid:15) > Theorem 3.1.
There exist n -dimensional reflections X , Z , . . . , X m , Z m , for m = e Ω( n(cid:15) ) , suchthat { X j , Z j } = 0 and (cid:107) [ S i , T j ] (cid:107) = O ( (cid:15) ) for all i (cid:54) = j and S, T ∈ {
X, Z } .Proof. By the Johnson-Lindenstrauss Lemma [JL84, DG03], e n(cid:15) / unit vectors can be chosen in R n so that for any pair | u (cid:105) , | v (cid:105) , |(cid:104) u | v (cid:105)| ≤ (cid:15) . Collecting these vectors in triples, we obtain m = e n(cid:15) / three-dimensional subspaces with the angles between any two in the range [ π − O ( (cid:15) ) , π ]. Let {| e j (cid:105) , | f j (cid:105) , | g j (cid:105)} , for j ∈ [ m ], be orthonormal bases for the subspaces.Let C , . . . , C n denote a 2 n -dimensional representation of the Clifford algebra, i.e., Hermitianmatrices that satisfy { C i , C j } = 2 δ ij . For each j ∈ [ m ], let E j = (cid:88) k (cid:104) k | e j (cid:105) C k F j = (cid:88) k (cid:104) k | f j (cid:105) C k G j = (cid:88) k (cid:104) k | g j (cid:105) C k . Then it is easy to check that for distinct
S, T ∈ {
E, F, G } , { S j , T j } = 0 and (cid:107){ S i , T j }(cid:107) = O ( (cid:15) )for i (cid:54) = j . Let X j = iE j F j and Z j = iE j G j ; these matrices are Hermitian, square to , andanti-commute. Moreover, for i (cid:54) = j and S, T ∈ {
X, Z } , we have (cid:107) [ S i , T j ] (cid:107) = O ( (cid:15) ).Appendix A gives an alternative proof of Theorem 3.1 using the exterior algebra. We provide two different methods for creating independent qubits from partially overlappingqubits. The first argument, given in Section 3.2.1, performs a careful analysis of a sequentialblock-diagonalization procedure. The second argument, in Section 3.2.3, is simpler but requires theintroduction of a larger Hilbert space in which to define the approximating operators.
We first consider the case of separating projections that nearly commute pairwise.
Theorem 3.2.
Let P , . . . , P n be projections on a finite-dimensional Hilbert space such that forsome (cid:15) ≤ n , (cid:107) [ P i , P j ] (cid:107) ≤ (cid:15) for all i, j .Then there exist projections Q , . . . , Q n with, for all i, j , [ Q i , Q j ] = 0 (cid:107) P i − Q i (cid:107) ≤ n(cid:15) . The bound in Theorem 3.2 is nearly tight; see Lemma 3.9 below.The proof of the theorem is constructive. It uses two basic operations, that we analyze with twolemmas. First we block-diagonalize operators with respect to a projection Q so that they commutewith Q . The first lemma bounds how block-diagonalizing two operators affects their commutator.6 emma 3.3. Let Q be a projection, and for operators P i , i = 1 , , let P (cid:48) i = QP i Q +( − Q ) P i ( − Q ) .Then [ Q, P (cid:48) i ] = 0 , (cid:107) P (cid:48) i − P i (cid:107) = (cid:107) [ Q, P i ] (cid:107) , and (cid:107) [ P (cid:48) , P (cid:48) ] (cid:107) ≤ (cid:107) [ P , P ] (cid:107) + 2 (cid:107) [ Q, P ] (cid:107) · (cid:107) [ Q, P ] (cid:107) . Proof.
Work in a basis in which Q is diagonal: Q = (
00 0 ). Then P i = (cid:0) A i B i C i D i (cid:1) and P (cid:48) i = (cid:0) A i D i (cid:1) .As [ Q, P i ] = (cid:0) B i − C i (cid:1) , (cid:107) P (cid:48) i − P i (cid:107) = max {(cid:107) B i (cid:107) , (cid:107) C i (cid:107)} = (cid:107) [ Q, P i ] (cid:107) . We also compute[ P , P ] = (cid:16) [ A ,A ]+ B C − B C A B + B D − A B − B D C A + D C − C A − D C [ D ,D ]+ C B − C B (cid:17) . Each diagonal block in [ P , P ] above, Q [ P , P ] Q and ( − Q )[ P , P ]( − Q ), must have norm atmost (cid:107) [ P , P ] (cid:107) . The claimed bound for (cid:107) [ P (cid:48) , P (cid:48) ] (cid:107) = max {(cid:107) [ A , A ] (cid:107) , (cid:107) [ D , D ] (cid:107)} follows.When one block-diagonalizes a projection, the result might not be a projection. The secondbasic operation consists in rounding the eigenvalues to the closest integer, 0 or 1. The second lemmabounds how this affects the commutator with another operator. Lemma 3.4.
Let Q be a projection and Q (cid:48) Hermitian with [ Q, Q (cid:48) ] = 0 and (cid:107) Q − Q (cid:48) (cid:107) < / . Thenfor any Hermitian P , (cid:107) [ Q, P ] (cid:107) ≤ (cid:107) [ Q (cid:48) , P ] (cid:107) − (cid:107) Q − Q (cid:48) (cid:107) . This bound can be much stronger than the trivial (cid:107) [ Q, P ] (cid:107) ≤ (cid:107) [ Q (cid:48) , P ] (cid:107) + 2 (cid:107) P (cid:107)(cid:107) Q − Q (cid:48) (cid:107) . It followsby substituting A = (cid:16) P (2 Q − )(2 Q − ) P (cid:17) , B = (cid:16) Q − ) PP (2 Q − ) 0 (cid:17) and Γ = | Q (cid:48) − | ⊕ | Q (cid:48) − | into the following theorem, and using | Q (cid:48) − | (2 Q − ) = (2 Q − ) | Q (cid:48) − | = 2 Q (cid:48) − . Theorem 3.5 ([BDK91, Theorem 1]) . If A and B are Hermitian, and Γ (cid:31) , then (cid:107) A − B (cid:107) ≤ (cid:107) Γ − (cid:107) · (cid:107) A Γ − Γ B (cid:107) . Proof of Theorem 3.2.
We proceed inductively. The induction hypothesis is that we have defined Q , . . . , Q k , P ( k ) k +1 , . . . , P ( k ) n such that • (cid:22) P ( k ) j (cid:22) , (cid:107) P ( k ) j − P j (cid:107) ≤ δ k , (cid:107) [ P ( k ) i , P ( k ) j ] (cid:107) ≤ (cid:15) k . • Q , . . . , Q k are projections, commuting with each other and all P ( k ) j , with (cid:107) P k − Q k (cid:107) ≤ δ k − .For the base case, δ = 0 and (cid:15) = (cid:15) .In the induction step, we let Q k +1 be the projection formed by rounding P ( k ) k +1 ’s eigenvalues to 0or 1, and define P ( k +1) k +2 , . . . , P ( k +1) n by block-diagonalizing the P ( k ) j operators with respect to Q k +1 : P ( k +1) j = Q k +1 P ( k ) j Q k +1 + ( − Q k +1 ) P ( k ) j ( − Q k +1 ) . For P (cid:23)
0, trivially (cid:107) [ Q, P ] (cid:107) ≤ (cid:107) [ Q (cid:48) , P ] (cid:107) + (cid:107) [ Q − Q (cid:48) , P − (cid:107) P (cid:107) ] (cid:107) ≤ (cid:107) [ Q (cid:48) , P ] (cid:107) + (cid:107) P (cid:107)(cid:107) Q − Q (cid:48) (cid:107) , but Lemma 3.4 isstill stronger. (cid:107) Q k +1 − P k +1 (cid:107) ≤ (cid:107) P ( k ) k +1 − P k +1 (cid:107) + (cid:107) Q k +1 − P ( k ) k +1 (cid:107) ≤ δ k . Also, 0 (cid:22) P ( k +1) j (cid:22) .Using Lemma 3.3, we compute (cid:107) P ( k +1) j − P j (cid:107) ≤ (cid:107) P ( k ) j − P j (cid:107) + (cid:107) P ( k +1) j − P ( k ) j (cid:107)≤ δ k + (cid:107) [ Q k +1 , P ( k ) j ] (cid:107)(cid:107) [ P ( k +1) i , P ( k +1) j ] (cid:107) ≤ (cid:107) [ P ( k ) i , P ( k ) j ] (cid:107) + 2 (cid:107) [ Q k +1 , P ( k ) i ] (cid:107) · (cid:107) [ Q k +1 , P ( k ) j ] (cid:107) . Thus we may take δ k +1 = δ k + max j (cid:107) [ Q k +1 , P ( k ) j ] (cid:107) and (cid:15) k +1 = (cid:15) k + 2 max j (cid:107) [ Q k +1 , P ( k ) j ] (cid:107) . Itremains to bound max j (cid:107) [ Q k +1 , P ( k ) j ] (cid:107) .The naive bound (cid:107) [ Q k +1 , P ( k ) j ] (cid:107) ≤ (cid:107) [ P ( k ) k +1 , P ( k ) j ] (cid:107) + 2 (cid:107) Q k +1 − P ( k ) k +1 (cid:107) ≤ (cid:15) k + 2 δ k is no good, as itallows the errors to grow exponentially with k . Instead, applying Lemma 3.4 gives (cid:13)(cid:13) [ Q k +1 , P ( k ) j ] (cid:13)(cid:13) ≤ (cid:15) k − δ k . Provided that all (cid:15) k ≤ (cid:15) and δ k ≤ /
4, (1 − δ k ) − ≤
2, and we obtain the recursions δ k +1 ≤ δ k + 2 (cid:15) k ≤ δ k + 4 (cid:15)(cid:15) k +1 ≤ (cid:15) k + 8 (cid:15) k ≤ (cid:15) k + 32 (cid:15) . Thus δ k +1 ≤ k + 1) (cid:15) and (cid:15) k +1 ≤ (cid:15) + 32 k(cid:15) . Given (cid:15) ≤ n , indeed (cid:15) k ≤ (cid:15) and δ k ≤ / The following theorem is an extension of Theorem 3.2 which allows us to separate (cid:15) -overlappingqubits.
Theorem 3.6.
Let X , Z , . . . , X n , Z n be Hermitian matrices each having eigenvalues in the range [ − , − (cid:15) ] ∪ [1 − (cid:15), , and satisfying (cid:107){ X j , Z j }(cid:107) ≤ (cid:15) and (cid:107) [ S i , T j ] (cid:107) ≤ (cid:15) for all i (cid:54) = j and S, T ∈ {
X, Z } .Assume (cid:15)/ (1 − (cid:15) ) ≤ n . Then there exist reflections X (cid:48) , Z (cid:48) , . . . , X (cid:48) n , Z (cid:48) n with { X (cid:48) j , Z (cid:48) j } = 0 , and [ S (cid:48) i , T (cid:48) j ] = 0 and (cid:107) S (cid:48) j − S j (cid:107) ≤ n(cid:15)/ (1 − (cid:15) ) + (cid:15) for all i (cid:54) = j and S, T ∈ {
X, Z } .Proof. Let H be the finite-dimensional Hilbert space on which the matrices act. Introduce n additional qubits, and on ( C ) ⊗ n ⊗ H , define R (cid:48) j − = σ xj ⊗ X j R (cid:48) j = σ zj ⊗ Z j , for j = 1 , . . . , n , where σ xj and σ zj are the standard Pauli operators acting on the j th added qubit.For Pauli operators σ and τ ,[ σ ⊗ A, τ ⊗ B ] = (cid:40) ( στ ) ⊗ [ A, B ] if [ σ, τ ] = 0( στ ) ⊗ { A, B } if { σ, τ } = 0 .Thus for all i, j , (cid:107) [ R (cid:48) i , R (cid:48) j ] (cid:107) ≤ (cid:15) . R , . . . , R n by rounding to ± R (cid:48) , . . . , R (cid:48) n . Theoperators R j still have the form (Pauli) ⊗ (Reflection). By Theorem 3.5, (cid:107) [ R i , R j ] (cid:107) ≤ − (cid:15) ) (cid:15) . Define projections P , . . . , P n by P j = ( + R j ). Then (cid:107) [ P i , P j ] (cid:107) = (cid:107) [ R i , R j ] (cid:107)≤
14 1(1 − (cid:15) ) (cid:15) . Applying Theorem 3.2 for separating projections yields projections Q , . . . , Q n satisfying[ Q i , Q j ] = 0 and (cid:107) Q j − P j (cid:107) ≤ · (2 n ) ·
14 1(1 − (cid:15) ) (cid:15) = 4 n(cid:15) (1 − (cid:15) ) , provided that (cid:15)/ (1 − (cid:15) ) ≤ / (64 n ).We claim that the reflections 2 Q j − − and 2 Q j − still have the form σ xj ⊗ X (cid:48) j and σ zj ⊗ Z (cid:48) j ,resepectively, for reflections X (cid:48) j and Z (cid:48) j on H . Indeed, the proof of the projections separationtheorem, Theorem 3.2, involved two basic operations:1. Block-diagonalizing an operator A with respect to a reflection R : A → ( + R ) A ( + R ) + ( − R ) A ( − R )= 12 ( A + RAR ) .
2. Rounding the eigenvalues of a Hermitian operator A to ± A = σ ⊗ A (cid:48) for a Pauli σ , and R = τ ⊗ R (cid:48) for a Pauli τ , then both of these basicoperations result in an operator σ ⊗ A (cid:48)(cid:48) , for the same Pauli σ .Thus indeed { X (cid:48) j , Z (cid:48) j } = 0 and [ S (cid:48) i , T (cid:48) j ] = 0 for i (cid:54) = j and S, T ∈ {
X, Z } . Also (cid:107) Q j − P j (cid:107) ≤ n(cid:15)/ (1 − (cid:15) ) implies (cid:107) S (cid:48) j − S j (cid:107) ≤ (cid:107) Q j − P j (cid:107) + (cid:107) R (cid:48) j − R j (cid:107)≤ n(cid:15) (1 − (cid:15) ) + (cid:15) . Since Theorem 3.6 yields n qubits in tensor product, the dimension of the ambient space H must be at least 2 n . Rephrasing this, we obtain: Corollary 3.7. In n dimensions, at most n qubits can be placed with pairwise “overlaps” (cid:107) [ S i , T j ] (cid:107) ≤ (cid:15) , if (cid:15)/ (1 − (cid:15) ) ≤ / (64 n ) . .2.3 SWAP-based argument If we are willing to work in a larger space, then there is a simpler argument for moving overlappingqubits into tensor product. Instead of repeatedly block-diagonalizing operators and rounding theireigenvalues to ±
1, as in Theorem 3.6, we can swap in fresh qubits to enforce a tensor-productstructure. We will show:
Theorem 3.8.
Let X , Z , . . . , X n , Z n be reflections on H , satisfying { X j , Z j } = 0 and (cid:107) [ S i , T j ] (cid:107) ≤ (cid:15) for all i (cid:54) = j and S, T ∈ {
X, Z } . Extend these operators by the identity to act on H ⊗ ( C ) ⊗ n .Then there exist reflections X (cid:48) , Z (cid:48) , . . . , X (cid:48) n , Z (cid:48) n on H ⊗ ( C ) ⊗ n , with { X (cid:48) j , Z (cid:48) j } = 0 , [ S (cid:48) i , T (cid:48) j ] = 0 and (cid:107) S (cid:48) j − S j (cid:107) ≤ n(cid:15) .Proof. For j ∈ [ n ], let S j = (cid:0) ⊗ + X j ⊗ σ xj + Z j ⊗ σ zj + i ( X j Z j ) ⊗ σ yj (cid:1) . Acting on H ⊗ ( C ) ⊗ n , S j swaps the j th added C register with the qubit defined by X j , Z j .For T ∈ { X, Z } and i ∈ { , . . . , j } define T ( i ) j = ( S · · · S i − ) T j ( S i − · · · S ) . Let T (cid:48) j = T ( j ) j = ( S · · · S j − ) T j ( S j − · · · S ).Then for i < j , (cid:107) [ S (cid:48) i , T (cid:48) j ] (cid:107) = (cid:107) [ S i , S i · · · S j − T j S j − · · · S i ] (cid:107) . This is 0, since for any operator A that is the identity on the i th added C register, [ S i , S i A S i ] = 0.Furthermore, (cid:107) T (cid:48) j − T j (cid:107) ≤ j − (cid:88) i =1 (cid:107) T ( i +1) j − T ( i ) j (cid:107) = j − (cid:88) i =1 (cid:107)S i T j S i − T j (cid:107) = j − (cid:88) i =1 (cid:107) [ S i , T j ] (cid:107)≤ j − (cid:88) i =1 (cid:0) (cid:107) [ X i , T j ] (cid:107) + (cid:107) [ Z i , T j ] (cid:107) + (cid:107) [ X i Z i , T j ] (cid:107) (cid:1) ≤ (cid:15) ( j − . Since Theorem 3.8 works in the larger space
H ⊗ ( C ) ⊗ n , unlike Theorem 3.6 it does not givean upper bound on the number of nearly independent qubits that can be packed into H . Ω( n(cid:15) ) movement is necessary Theorem 3.6 shows that n qubits with pairwise “overlaps” at most (cid:15) can be separated into tensorproduct by moving each qubit O ( n(cid:15) ) in operator norm. Is the loss of a factor of n necessary? Thefollowing example shows that our bound is essentially tight.10 emma 3.9. For any integer n , and any (cid:15) ∈ [0 , π/n ] , there exist n qubits X , Z , . . . , X n , Z n in ( C ) ⊗ (2 n ) such that (cid:107) [ S i , T j ] (cid:107) ≤ (cid:15) for all i (cid:54) = j and S, T ∈ {
X, Z } but such that for any independent qubits X (cid:48) , Z (cid:48) , . . . , X (cid:48) n , Z (cid:48) n (with [ S (cid:48) i , T (cid:48) j ] = 0 for i (cid:54) = j ), max ≤ j ≤ nS ∈{ X,Z } (cid:13)(cid:13) S j − S (cid:48) j (cid:13)(cid:13) ≥ n(cid:15) π . Proof.
Construct qubits X j , Z j as the standard qubits, except with the second n qubit operatorsperturbed by the Hamiltonian H = ( σ z + · · · + σ zn )( σ zn +1 + · · · + σ z n ) . That is, X j = σ xj , Z j = σ zj for j ≤ n , and X j = e i(cid:15)H σ xj e − i(cid:15)H , Z j = e i(cid:15)H σ zj e − i(cid:15)H = σ zj for j > n .Then if j, k ≤ n or j, k > n , the operators for qubits j and k commute. If j ≤ n < k , thenthe operators for qubits j and k commute, except for X j and X k . We compute (cid:107) [ X j , X k ] (cid:107) = (cid:107) X j X k X j − X k (cid:107) = (cid:107) e − i(cid:15)H σ xj e i(cid:15)H σ xk e − i(cid:15)H σ xj e i(cid:15)H − σ xk (cid:107) = (cid:107) e i(cid:15)σ zj σ zk − (cid:107) = | e i(cid:15) − | ≤ (cid:15) .Let X (cid:48) , . . . , X (cid:48) n be any pairwise commuting reflections. Let J = { , . . . , n } , K = { n + 1 , . . . , n } .Let X J = (cid:81) j ∈ J X j , X K = (cid:81) k ∈ K X k . Similarly define X (cid:48) J , X (cid:48) K and σ xJ , σ xK . Thus X J = σ xJ , X K = e i(cid:15)H σ xK e − i(cid:15)H . In order to lower-bound max j (cid:107) X j − X (cid:48) j (cid:107) , we study (cid:107) [ X J , X K ] (cid:107) = (cid:107) ( X J X K ) − (cid:107) .On one hand, since the X (cid:48) j operators commute, ( X (cid:48) J X (cid:48) K ) = . By triangle inequalities, andusing (cid:107) X j (cid:107) = (cid:107) X (cid:48) j (cid:107) = 1 for all j , (cid:107) X J X K − X (cid:48) J X (cid:48) K (cid:107) ≤ (cid:80) j (cid:107) X j − X (cid:48) j (cid:107) , and hence (cid:107) ( X J X K ) − (cid:107) ≤ (cid:88) j (cid:107) X (cid:48) j − X j (cid:107) ≤ n · max j (cid:107) X (cid:48) j − X j (cid:107) . (1)On the other hand, ( X J X K ) = σ xJ (cid:0) e i(cid:15)H σ xK e − i(cid:15)H (cid:1) σ xJ (cid:0) e i(cid:15)H σ xK e − i(cid:15)H (cid:1) = e − i(cid:15)H σ xK e i(cid:15)H σ xK e − i(cid:15)H = e − i(cid:15)H . Since (cid:107) H (cid:107) = n /
4, provided that n (cid:15) ≤ π it holds that (cid:13)(cid:13) ( X J X K ) − (cid:13)(cid:13) = | e in (cid:15) − | ≥ π · n (cid:15) . (2)Combining the bounds (1) and (2) gives π n (cid:15) ≤ (cid:107) ( X J X K ) − (cid:107) ≤ n · max j (cid:107) X (cid:48) j − X j (cid:107) , ormax j (cid:107) X (cid:48) j − X j (cid:107) ≥ n(cid:15)/ (2 π ). A problem with both Theorem 3.6 and Theorem 3.8 is that they might be difficult to apply to realexperimental systems. This is because it is difficult to establish the assumption of qubits nearly intensor product, (cid:107) [ S i , T j ] (cid:107) ≤ (cid:15) for i (cid:54) = j and S, T ∈ {
X, Z } . In addition to the operators, a physicalsystem involves an underlying state | ψ (cid:105) . The operators can be understood only in terms of theireffects on | ψ (cid:105) . Consider for example a Hilbert space that splits as H ⊕ H (cid:48) , where | ψ (cid:105) is supportedonly on H and available operators leave H invariant. Then there is no experimental way to fathom11he operators’ behavior, e.g., their commutation relationships, on H (cid:48) . Theorems 3.6 and 3.8 cannotbe applied. This example might not seem so troubling, because we can simply restrict everythingto H ; but it becomes more problematic if | ψ (cid:105) , say, has nonzero but very small support on H (cid:48) .We would like qubit-separation theorems that have experimentally accessible assumptions. Inparticular, the theorems’ assumptions should be stated relative to the system’s state | ψ (cid:105) . Forexample, in Theorems 3.6 and 3.8 we might loosen the assumption (cid:107) [ S i , T j ] (cid:107) ≤ (cid:15) for i (cid:54) = j to beonly (cid:107) [ S i , T j ] | ψ (cid:105)(cid:107) ≤ (cid:15) . Naturally, the conclusions will have to be correspondingly weakened. In theabove example with H ⊕ H (cid:48) , if the reflections are far from commuting on H (cid:48) then we cannot hopeto find nearby commuting operators, (cid:107) S (cid:48) j − S j (cid:107) ≈
0; but perhaps we can get (cid:107) ( S (cid:48) j − S j ) | ψ (cid:105)(cid:107) ≈ S and T , are close to commuting on a state | ψ (cid:105) : [ S, T ] | ψ (cid:105) ≈
0. The protocol is very simple:measure S , measure T , then measure S again. If S and T commute on | ψ (cid:105) , then the two S measurements will give the same result; and, intuitively, when they do not commute measuring T will disturb the state and make it less likely to get the same S result.2. However, in Section 4.2, we show that the condition [ S i , T j ] | ψ (cid:105) ≈ X (cid:48) , Z (cid:48) , . . . , X (cid:48) n , Z (cid:48) n .In fact, we give an explicit construction of a state | ψ (cid:105) and n qubit operators X , Z , . . . , X n , Z n in < n dimensions such that for i (cid:54) = j , [ S i , T j ] | ψ (cid:105) = 0 precisely. Since n ≤ n for n ≥
4, thedimension of the space is not sufficient to fit n independent qubits.(We also show why the basic induction argument used to prove Theorem 3.6 fails whenerrors are measured relative to a state | ψ (cid:105) . The errors accumulate too rapidly, leading to anexponential dependence on n , instead of polynomial.)3. We remedy this problem in Section 4.3 with a more advanced testing protocol. Intuitively, theimproved protocol tests not just pairwise commutation relationships, such as S i T j | ψ (cid:105) ≈ T j S i | ψ (cid:105) ,but also higher-order relationships such as S i T j U k | ψ (cid:105) ≈ U k T j S i | ψ (cid:105) . The protocol is still quitesimple, though. Basically, measure all the qubit operators in order (either X , Z , X , Z , . . . or Z , X , Z , X , . . . ), then go back and measure a random qubit operator ( Z j or X j , respectively),and verify that the measurement result is unchanged. We show that if the protocol acceptswith probability 1 − (cid:15) , then the qubit operators “simulate” n independent qubit operatorsin a certain sense. In particular, as a corollary, the system’s dimension must be at least(1 − O ( n (cid:15) ))2 n .The dimension bound is not fully satisfactory. A 2 n lower bound would be preferable. However,speculatively, the simulation statement might be strong enough to form the foundation for ananalysis that the system can be used as an n -qubit quantum computer. Such an extension isnontrivial, though, and we leave it to future work. We present a protocol that can be used to test whether two reflections approximately commute on agiven state. 12 heorem 4.1.
Let S and T be reflections, acting on a state | ψ (cid:105) . Consider the following protocol:1. Measure S .2. Measure T , but ignore the result.3. Measure S again. Accept if the result is unchanged.Then the probability of accepting is given by Pr[accept] = 1 − (cid:13)(cid:13) [ S, T ] | ψ (cid:105) (cid:13)(cid:13) . Proof.
For a, b ∈ { , } , let S a = ( + ( − a S ) and T b = ( + ( − b T ). Then since [ S, T ] = − [ S, T ] = [ S, T ], (cid:107) [ S, T ] | Ψ (cid:105)(cid:107) = 2 (cid:0) (cid:107) [ S, T ] | ψ (cid:105)(cid:107) + (cid:107) [ S, T ] | ψ (cid:105)(cid:107) (cid:1) = (cid:88) a,b (cid:13)(cid:13) S a [ S, T b ] | ψ (cid:105) (cid:13)(cid:13) , where we have used (cid:107)| φ (cid:105)(cid:107) = (cid:107) S | φ (cid:105)(cid:107) + (cid:107) S | φ (cid:105)(cid:107) for any | φ (cid:105) . Then from S a S = SS a = ( − a S a ,we find S a [ S, T b ] = S a [ S, T b ]( S + S ) = 2( − a S a T S ¯ a , so (cid:107) [ S, T ] | ψ (cid:105)(cid:107) = 8 (cid:88) a,b (cid:13)(cid:13) S a T b S ¯ a | ψ (cid:105) (cid:13)(cid:13) = 8 (1 − Pr[accept]) . In the projection separating argument of Theorem 3.2, the key observation was that for projections P , Q , R with (cid:107) [ P, Q ] (cid:107) , (cid:107) [ P, R ] (cid:107) ≤ δ and (cid:107) [ Q, R ] (cid:107) ≤ (cid:15) , if Q and R are both block-diagonalized withrespect to P then the results still nearly commute: (cid:13)(cid:13)(cid:2) P QP + ( − P ) Q ( − P ) , P RP + ( − P ) R ( − P ) (cid:3)(cid:13)(cid:13) ≤ (cid:15) + 2 δ . The quadratic dependence on δ meant that errors did not accumulate badly through the induction.Here is a counterexample showing that errors can accumulate badly in block diagonalization ifwe measure errors relative to a state | ψ (cid:105) , using (cid:107) [ P, Q ] | ψ (cid:105)(cid:107) . Define P , Q , R and | ψ (cid:105) as P = (cid:32) δ / / / / δ (cid:33) Q = (cid:18) (cid:19) R = (cid:32) / /
20 0 1 / / (cid:33) | ψ (cid:105) = (cid:18) (cid:19) . (3)Then P , Q and R are projections (up to second order in δ for P ), with (cid:107) [ P, Q ] | ψ (cid:105)(cid:107) , (cid:107) [ P, R ] | ψ (cid:105)(cid:107) = O ( δ ),[ Q, R ] | ψ (cid:105) = 0, and yet (cid:13)(cid:13)(cid:2) P QP + ( − P ) Q ( − P ) , P RP + ( − P ) R ( − P ) (cid:3) | ψ (cid:105) (cid:13)(cid:13) = Ω( δ ) . The idea is that Q and R commute on the first two dimensions, and are far from commuting on thelast two dimensions; but this property is broken by the block diagonalization.13his example suggests that in a simple induction argument, starting with projections P , . . . , P n having pairwise commutators (cid:107) [ P i , P j ] | ψ (cid:105)(cid:107) ∼ (cid:15) , after block-diagonalizing with respect to P , theerrors can grow to ∼ (cid:15) , then to ∼ (cid:15) after block-diagonalizing with respect to the new P , and soon; the errors potentially grow exponentially.In fact, it is not only our proof of Theorems 3.2 and 3.6 that fails when errors are measuredrelative to a state | ψ (cid:105) . The theorems themselves fail, as shown by the following construction. Lemma 4.2.
For any n and k ∈ [ n ] , there exists a space H of dimension at most (cid:80) kj =0 (cid:0) nj (cid:1) , avector | ψ (cid:105) ∈ H and n qubits X j , Z j such that S (1) j · · · S ( k ) j k | ψ (cid:105) = S σ (1) j σ (1) · · · S σ ( k ) j σ ( k ) | ψ (cid:105) for all distinct indices j , . . . , j k ∈ [ n ] , S (1) , . . . , S ( k ) ∈ { X, Z } , and permutations σ of [ k ] . In particular, for k = 2, the lemma places n qubits in O ( n ) dimensions—for example, fourqubits in 12 dimensions—such that [ S i , T j ] | ψ (cid:105) = 0 for all i (cid:54) = j and S, T ∈ {
X, Z } . Proof.
Let us begin by explaining the n = 4, k = 2 special case of the construction. Define H to have orthonormal basis | (cid:105) , | (cid:105) , . . . , | (cid:105) , | (cid:105) , . . . , | (cid:105) , | d (cid:105) , i.e., all n -bit strings ofHamming weight at most k , together with an additional vector | d (cid:105) . Let | ψ (cid:105) = | (cid:105) , and considerthe following operators for the first qubit: X = d d Z = d –1 –1 –1 –1 –1 d –1 . Unspecified matrix entries are 0. X and Z can be obtained from X and Z by switching the firstand second bits in each basis element, leaving | d (cid:105) alone; and similarly for X , Z and X , Z . Then P i = , { X i , Z i } = 0 and [ P i , Q j ] | ψ (cid:105) = 0, for i (cid:54) = j and P, Q ∈ {
X, Z } .The idea behind this construction is that X j , Z j act largely as the Pauli operators σ xj , σ zj .However, we have truncated the standard basis | (cid:105) , . . . , | (cid:105) for ( C ) ⊗ to include only stringsof Hamming weight ≤
2. Since applying σ x to | (cid:105) , | (cid:105) and | (cid:105) would give strings ofHamming weight 3, we instead pair these dimensions up arbitrarily for X , and define Z on themto make it anti-commute with X . The extra dimension | d (cid:105) is needed to make the total dimensioneven.It is straightforward to generalize the example: by truncating strings at Hamming weight k the same construction places n qubits in (cid:80) kj =0 (cid:0) nj (cid:1) dimensions (or one more if this dimension isodd), such that any combination of up to k qubit operators commute on | ψ (cid:105) = | n (cid:105) , e.g., if k ≥ X X X | ψ (cid:105) = X X X | ψ (cid:105) = | n − (cid:105) . n independent qubits The problem with the protocol in Theorem 4.1 is that it only tests commutation between pairs ofoperators on the state | ψ (cid:105) : [ S, T ] | ψ (cid:105) ≈
0. Lemma 4.2 shows that n qubits in only O ( n ) dimensions14 rotocol to test for n independent qubits Let | ψ (cid:105) ∈ H be a state. Let X , Z , . . . , X n , Z n be qubit operators on H , i.e., reflections satisfying { X j , Z j } = 0 for all j .1. With equal probabilities 1 /
2, measure the reflections in order, either X , Z , X , Z , . . . , or Z , X , Z , X , . . . .2. Pick a uniformly random index j ∈ [ n ]. If Z went second in step (1), then measure Z j ; andif X went second, then measure X j . Accept if the result is unchanged from the operator’sprevious measurement. Otherwise reject.Figure 2: Protocol to test for n independent qubits.can pass this test on every pair. The lemma furthermore suggests that any test involving qubitoperator sequences of length o ( n ) can be satisfied in dimension 2 o ( n ) . Therefore, we need a protocolthat has at least n steps.Figure 2 gives our testing protocol. We argue that if the protocol accepts with high probability,then the n overlapping qubits X j , Z j are nearly equivalent to n independent qubits ˆ X j , ˆ Z j in anenlarged space H (cid:48) = H ⊗ ( C ) ⊗ n . Theorem 4.3.
Consider the protocol of Figure 2. Assume the probability it accepts is at least − (cid:15) .Let | EPR (cid:105) = √ ( | (cid:105) + | (cid:105) ) . Let | Ψ (cid:105) = | ψ (cid:105) ⊗ | EPR (cid:105) ⊗ n ∈ H (cid:48) = H ⊗ ( C ) ⊗ n , and let | Ψ (cid:105) beobtained from | Ψ (cid:105) by swapping each qubit X j , Z j with the first half of one of the EPR states, in order j = 1 , . . . , n . (See Figure 3.) Then there exist n independent qubits, given by ˆ X , ˆ Z , . . . , ˆ X n , ˆ Z n ,on H (cid:48) such that for any sequence of qubit operators U j , . . . , U j k , where U j acts on the X j , Z j qubitand (cid:107) U j (cid:107) ≤ , (cid:13)(cid:13) U j · · · U j k | Ψ (cid:105) − ˆ U j · · · ˆ U j k | Ψ (cid:105) (cid:13)(cid:13) = O ( kn √ (cid:15) ) . (4) Here ˆ U j is the same operator as U j , except acting on the ˆ X j , ˆ Z j qubit. That is, if U j has Pauliexpansion U j = α j + β j X j + γ j Z j + δ j ( iX j Z j ) for scalars α j , β j , γ j , δ j , then ˆ U j = α j + β j ˆ X j + γ j ˆ Z j + δ j ( i ˆ X j ˆ Z j ) . Observe that if the X j , Z j qubits are independent of each other, then the measurements ondifferent qubits commute, and so the protocol accepts with probability one. In that case, there isnothing to show. In general, however, measuring qubits j + 1 , . . . , n can disturb the last measurementon qubit j .The EPR state appears in the conclusion of Theorem 4.3 even though it is not used in thetesting protocol. Essentially this is because of the following two properties of | EPR (cid:105) :1. Depolarizing a qubit, i.e., replacing it with the maximally mixed state, is equivalent toswapping it with the first qubit of a fresh EPR state then tracing out the EPR state’s registers.2. For any 2 × M , ( I ⊗ M ) | EPR (cid:105) = ( M T ⊗ I ) | EPR (cid:105) .The second property is key in our analysis for algebraically manipulating operators to showapproximate commutation. To see how, consider for example a state | φ (cid:105) that involves four qubits,15igure 3: The state | Ψ (cid:105) is given by | ψ (cid:105) ⊗ | EPR (cid:105) ⊗ n , where the EPR states are on qubits 1 (cid:48) and1 (cid:48)(cid:48) , 2 (cid:48) and 2 (cid:48)(cid:48) , and so on. To get | Ψ (cid:105) , swap qubit j (cid:48) with the qubit in H defined by X j , Z j , for j = 1 , . . . , n . Observe that starting from | ψ (cid:105) and depolarizing the X j , Z j qubits, for j = 1 , . . . , n , isequivalent to tracing out all j (cid:48) and j (cid:48)(cid:48) qubits from | Ψ (cid:105)(cid:104) Ψ | .labeled 1 , , (cid:48) , (cid:48) , where the j (cid:48) qubits do not overlap with any others. If | φ (cid:105) is close to an EPR stateon qubits (1 , (cid:48) ) and (2 , (cid:48) ), then operators on qubits 1 and 2 necessarily nearly commute on | φ (cid:105) : U V | φ (cid:105) ≈ U V T (cid:48) | φ (cid:105) = V T (cid:48) U | φ (cid:105)≈ V T (cid:48) U T (cid:48) | φ (cid:105) = U T (cid:48) V T (cid:48) | φ (cid:105)≈ U T (cid:48) V | φ (cid:105) = V U T (cid:48) | φ (cid:105)≈ V U | φ (cid:105) . The trick is to pull operators from one side of an approximate EPR state to the other, commutethem there, then pull them back.
Proof of Theorem 4.3.
To analyze the protocol, we relate it to a separate protocol that is basedon swapping qubits with halves of EPR states. Observe that measuring either X i then Z i , or Z i then X i , and discarding the second measurement result, is equivalent to depolarizing the qubit.Depolarizing a qubit is equivalent to swapping it with one half of | EPR (cid:105) and tracing out the originalEPR state’s registers. Therefore, the protocol of Figure 2 accepts with the same probability as thefollowing protocol:1. Append to the system n EPR states, on qubits labeled 1 (cid:48) , (cid:48)(cid:48) , . . . , n (cid:48) , n (cid:48)(cid:48) . Thus the system is inthe state | Ψ (cid:105) = | ψ (cid:105) ⊗ | EPR (cid:105) ⊗ n ∈ H ⊗ ( C (cid:48) ⊗ C (cid:48)(cid:48) ) ⊗ · · · ⊗ ( C n (cid:48) ⊗ C n (cid:48)(cid:48) ); see Figure 3.2. For i from 1 up to n , swap the qubit defined by X i , Z i with the new qubit i (cid:48) .3. Pick a uniformly random index j ∈ [ n ]. With equal probabilities 1 /
2, measure either X j and σ xj (cid:48)(cid:48) , or Z j and σ zj (cid:48)(cid:48) . Accept if the measurement results are the same, both +1 or both − α ∈ { x, z } , measuring σ αj (cid:48)(cid:48) at the end of the protocol is equivalent to measuring σ αj (cid:48) atthe start, which is also equivalent to measuring just after swapping with the X j , Z j qubit.If the protocol accepts with probability 1 − (cid:15) , then for probabilities (cid:15) j satisfying (cid:15) = n (cid:80) j (cid:15) j , wehave min (cid:8) (cid:107) ( + X j ⊗ σ xj (cid:48)(cid:48) ) | Ψ (cid:105)(cid:107) , (cid:107) ( + Z j ⊗ σ zj (cid:48)(cid:48) ) | Ψ (cid:105)(cid:107) (cid:9) ≥ − (cid:15) j , where | Ψ (cid:105) is the state afterthe swap gates in step (2). In particular,max (cid:110)(cid:13)(cid:13) X j ⊗ σ xj (cid:48)(cid:48) | Ψ (cid:105) − | Ψ (cid:105) (cid:13)(cid:13) , (cid:13)(cid:13) Z j ⊗ σ zj (cid:48)(cid:48) | Ψ (cid:105) − | Ψ (cid:105) (cid:13)(cid:13)(cid:111) ≤ (cid:112) (cid:15) j . This implies that for any one-qubit operator U j acting on the X j , Z j qubit, U j | Ψ (cid:105) ≈ U Tj (cid:48)(cid:48) | Ψ (cid:105) , where U j (cid:48)(cid:48) is the same operator, but acting on the j (cid:48)(cid:48) qubit. More precisely, if U j = α j + β j X j + γ j Z j +16 j ( iX j Z j ) for complex scalars α j , β j , γ j , δ j , then U Tj (cid:48)(cid:48) = α j + β j σ xj (cid:48)(cid:48) + γ j σ zj (cid:48)(cid:48) − δ j σ yj (cid:48)(cid:48) ; and, sincemax {| α j | , | β j | , | γ j | , | δ j |} ≤ (cid:107) U j (cid:107) , (cid:13)(cid:13) ( U j − U Tj (cid:48)(cid:48) ) | Ψ (cid:105) (cid:13)(cid:13) ≤ ( | β j | + | γ j | + 2 | δ j | ) · (cid:112) (cid:15) j ≤ (cid:107) U j (cid:107) · (cid:112) (cid:15) j . For each i , let S i be the operator on that swaps the X i , Z i qubit with the new qubit i (cid:48) : S i = (cid:0) + X i ⊗ σ xi (cid:48) + Z i ⊗ σ zi (cid:48) + i ( X i Z i ) ⊗ σ yi (cid:48) (cid:1) . For i ≤ j , let S i,j = S i S i +1 . . . S j and S j,i = S j S j − . . . S i .Thus | Ψ (cid:105) = S n, | Ψ (cid:105) .Let ˆ P i = S n,i +1 P i S i +1 ,n = S n,i σ Pi (cid:48) S i,n = S n, σ Pi (cid:48) S ,n . As [ σ Pi (cid:48) , σ Qj (cid:48) ] = 0 for i (cid:54) = j and P, Q ∈ {
X, Z } ,so too [ ˆ P i , ˆ Q j ] = 0.Observe that ˆ U j | Ψ (cid:105) = U Tj (cid:48)(cid:48) | Ψ (cid:105) , (5)since ˆ U j S n, | Ψ (cid:105) = ( S n, U j (cid:48) S ,n ) S n, | Ψ (cid:105) = S n, U j (cid:48) | Ψ (cid:105) = S n, U Tj (cid:48)(cid:48) | Ψ (cid:105) , where the last equality is because | Ψ (cid:105) includes an EPR state between qubits j (cid:48) and j (cid:48)(cid:48) . It followsthat for any unitary U acting only on the X j , Z j qubit, (cid:13)(cid:13) ( U j − ˆ U j ) | Ψ (cid:105) (cid:13)(cid:13) ≤ (cid:112) (cid:15) j . (6)Now consider a sequence of operators U j , . . . , U j k , where U j acts on the X j , Z j qubit and (cid:107) U j (cid:107) ≤ U j | Ψ (cid:105) = U Tj (cid:48)(cid:48) | Ψ (cid:105) givesˆ U j · · · ˆ U j k | Ψ (cid:105) = ˆ U j · · · ˆ U j k − U Tj (cid:48)(cid:48) k | Ψ (cid:105) = U Tj (cid:48)(cid:48) k ˆ U j · · · ˆ U j k − | Ψ (cid:105) = · · · = U Tj (cid:48)(cid:48) k · · · U Tj (cid:48)(cid:48) | Ψ (cid:105) . To continue, iterate on U j | Ψ (cid:105) ≈ U Tj (cid:48)(cid:48) | Ψ (cid:105) : ≈ U j U Tj (cid:48)(cid:48) k · · · U Tj (cid:48)(cid:48) | Ψ (cid:105)≈ · · ·≈ U j · · · U j k | Ψ (cid:105) . The overall error satisfies (cid:13)(cid:13) U j · · · U j k | Ψ (cid:105) − ˆ U j · · · ˆ U j k | Ψ (cid:105) (cid:13)(cid:13) ≤ k · (cid:107) U j (cid:96) (cid:107) · (cid:112) (cid:15) j (cid:96) = O ( k √ n(cid:15) ) . In Theorem 4.3, the definition of | Ψ (cid:105) requires adding to H an additional ancilla register ( C ) ⊗ n .It is therefore not clear that the theorem should imply an exponential lower bound on the dimensionof H . In fact, though, it does lower-bound dim H :17 orollary 4.4. If the protocol in Figure 2 accepts with probability at least − (cid:15) , then dim H ≥ (cid:0) − O ( n (cid:15) ) (cid:1) n . Proof.
For ( a, b ) ∈ { , } n × { , } n let | Ψ a,b (cid:105) = (cid:0) X a n n Z b n n (cid:1) · · · (cid:0) X a Z b (cid:1) | Ψ (cid:105) . Claim 4.5.
The | Ψ a,b (cid:105) satisfy dim Span {| Ψ a,b (cid:105)} ≥ (cid:0) − O ( n (cid:15) ) (cid:1) n .Proof. Let B = (cid:80) a,b | Ψ a,b (cid:105)(cid:104) a, b | . Adopt the notation from the proof of Theorem 4.3. For k ∈{ , . . . , n } define | ˆΨ ( k ) a,b (cid:105) similarly to | Ψ a,b (cid:105) , except using the operators ˆ X j and ˆ Z j in place of X j and Z j for j ≤ k . Thus | ˆΨ (0) a,b (cid:105) = | Ψ a,b (cid:105) . Let | ˆΨ a,b (cid:105) = | ˆΨ ( n ) a,b (cid:105) and define ˆ B as B using the | ˆΨ a,b (cid:105) instead of | Ψ a,b (cid:105) . Using the triangle inequality and (cid:107) X j (cid:107) , (cid:107) Z j (cid:107) ≤ (cid:13)(cid:13) | ˆΨ a,b (cid:105) − | Ψ a,b (cid:105) (cid:13)(cid:13) ≤ n (cid:88) k =1 (cid:13)(cid:13) | ˆΨ ( k ) a,b (cid:105) − | ˆΨ ( k − a,b (cid:105) (cid:13)(cid:13) ≤ n (cid:88) k =1 (cid:13)(cid:13)(cid:13)(cid:0) ˆ X a k k ˆ Z b k k − X a k k Z b k k (cid:1)(cid:16) (cid:89) j In Theorem 3.6, different qubits overlapping by (cid:15) = O (1 /n ) already implies dim H ≥ n . In contrast, in Corollary 4.4, (cid:15) must be exponentially small before dim H ≥ n is required. Isthis polynomial versus exponential separation a consequence of loose analysis, an inherent drawbackof the protocol in Figure 2, or an inherent property of any efficient state-dependent qubit testingprotocol?The following example suggests at least that our analysis is not too loose. Let H = Span {| x (cid:105) : x (cid:54) = 0 n , n } ⊂ ( C ) ⊗ n . Define n qubits by Z j = σ zj | H and X j = σ xj | H + σ xj ( | n (cid:105)(cid:104) n | + | n (cid:105)(cid:104) n | ) σ xj .That is, while σ xj maps the basis states σ xj | n (cid:105) and σ xj | n (cid:105) outside of H , X j instead maps them toeach other. Even though dim H = 2 n − < n , it seems that these n qubits can pass our testingprotocol with probability − / exp( n ) . Acknowledgements We would like to thank Greg Kuperberg for helpful comments, particularly regarding the proofof Theorem 3.1. R.C., B.R. and C.S. supported by NSF grant CCF-1254119 and ARO grantW911NF-12-1-0541. T.V. supported by NSF CAREER grant CCF-1553477, an AFOSR YIP award,and the IQIM, an NSF Physics Frontiers Center (NFS Grant PHY-1125565) with support of theGordon and Betty Moore Foundation (GBMF-12500028). References [AHSL15] Tameem Albash, Itay Hen, Federico M. Spedalieri, and Daniel A. Lidar. Reexaminationof the evidence for entanglement in a quantum annealer. Phys. Rev. A , 92:062328, 2015,arXiv:1506.3539 [quant-ph].[Alo03] Noga Alon. Problems and results in extremal combinatorics—I. Discrete Mathematics ,273(1-3):31–53, 2003. EuroComb’01.[BDK91] Rajendra Bhatia, Chandler Davis, and Fuad Kittaneh. Some inequalities for commutatorsand an application to spectral variation. Aequationes Mathematicae , 41(1):70–78, 1991.[BF91] L´aszl´o Babai and Katalin Friedl. Approximate representation theory of finite groups. In Proc. 32nd IEEE FOCS , pages 733–742, 1991.[CRSV16] Rui Chao, Ben W. Reichardt, Chris Sutherland, and Thomas Vidick. Test for a largeamount of entanglement, using few measurements. 2016, arXiv:1610.00771 [quant-ph]. A natural generalization of this construction removes all strings of Hamming weight < t or > n − t , with Z j = σ zj | H and X j | x (cid:105) = σ xj | x (cid:105) except X j | x (cid:105) = | x (cid:105) when σ xj | x (cid:105) would cross the boundary. We omit the details. Random Structures and Algorithms , 22(1):60–65, 2003.[JL84] William B. Johnson and Joram Lindenstrauss. Extensions of Lipschitz mappings intoa Hilbert space. In Conf. on Modern Analysis and Probability, 1982 , volume 26 of Contemporary Mathematics , pages 189–206. Amer. Math. Soc., Providence, RI, 1984.[KL78] G. A. Kabatjanski˘ı and V. I. Levenˇste˘ın. Bounds for packings on the sphere and in space. Problemy Peredaˇci Informacii , 14(1):3–25, 1978.[KTW14] Jedrzej Kaniewski, Marco Tomamichel, and Stephanie Wehner. Entropic uncertaintyfrom effective anticommutators. Physical Review A , 90(1):012332, 2014, arXiv:1402.5722[quant-ph].[Kup14] Greg Kuperberg. Personal communication, February 2014.[Lor93] Terry A. Loring. C ∗ -algebras generated by stable relations. J. Functional Analysis ,112(1):159–203, 1993.[MR15] Cristopher Moore and Alexander Russell. Approximate representations, approximatehomomorphisms, and low-dimensional embeddings of groups. SIAM Journal on DiscreteMathematics , 29(1):182–197, 2015.[MY98] Dominic Mayers and Andrew Yao. Quantum cryptography with imperfect apparatus. In Proc. 39th IEEE FOCS , pages 503–509, 1998, arXiv:quant-ph/9809039.[MYS12] Matthew McKague, Tzyh Haur Yang, and Valerio Scarani. Robust self-testing of thesinglet. J. Phys. A: Math. Theor. , 45:455304, 2012, arXiv:1203.2976 [quant-ph].[Nay99] Ashwin Nayak. Optimal lower bounds for quantum automata and random access codes.In Proc. 40th IEEE FOCS , pages 369–376, 1999, arXiv:quant-ph/9904093.[Tao13] Terence Tao. A cheap version of the Kabatjanskii-Levenstein bound for almost orthogonalvectors. July 2013. https://terrytao.wordpress.com/2013/07/18/a-cheap-version-of-the-kabatjanskii-levenstein-bound-for-almost-orthogonal-vectors/. A Qubit packing using the exterior algebra An alternative proof of Theorem 3.1 was suggested to the authors by Greg Kuperberg [Kup14]. Therough idea is to begin by packing nearly orthogonal unit vectors in R n , then define qubits usingfermion creation and annihilation operators on the 2 n -dimensional exterior algebra. Proof of Theorem 3.1. By the Johnson-Lindenstrauss Lemma [JL84, DG03], e n(cid:15) / unit vectors canbe chosen in R n so that for any pair | u (cid:105) , | v (cid:105) , |(cid:104) u | v (cid:105)| ≤ (cid:15) . Pairing these vectors up arbitrarily, weobtain m = e n(cid:15) / two-dimensional planes the angles between any two of which are in the range( π − (cid:15), π ]. 20f | (cid:105) , . . . , | n (cid:105) is a basis for R n , let Λ( R n ) be the 2 n -dimensional exterior algebra, with basis | i (cid:105) ∧ | i (cid:105) ∧ . . . ∧ | i k (cid:105) for i , . . . , i k ∈ [ n ] and k = 0 , , . . . , n . For a unit vector | v (cid:105) ∈ R n and | w (cid:105) ∈ Λ( R n ), define the fermion creation and annihilation operators a † v | w (cid:105) = | v (cid:105) ∧ | w (cid:105) a v | w (cid:105) = ( (cid:104) v | ⊗ ) | w (cid:105) . Observe that this definition is basis independent, in the sense that for any unitary R on R n , a † Rv ˆ R | w (cid:105) = ˆ Ra † v | w (cid:105) a Rv ˆ R | w (cid:105) = ˆ Ra v | w (cid:105) , where ˆ R ( | v (cid:105) ∧ · · · ∧ | v k (cid:105) ) = ( R | v (cid:105) ) ∧ · · · ∧ ( R | v k (cid:105) ).If we choose a basis for R n beginning with | v (cid:105) , then a † v a v projects onto those basis terms inΛ( R n ) that include | v (cid:105) , while a v a † v projects onto the complementary set of basis terms. Thus a † v a v + a v a † v = , while also a v = ( a † v ) = 0. Furthermore, if | u (cid:105) is a unit vector perpendicular to | v (cid:105) ,then the anticommutators satisfy { a v , a u } = { a † v , a † u } = 0, as | u (cid:105) ∧ | v (cid:105) = −| v (cid:105) ∧ | u (cid:105) , while if | w (cid:105) has k terms, a u a † v | w (cid:105) = ( (cid:104) u | ⊗ )( | v (cid:105) ∧ | w (cid:105) )= ( − k ( (cid:104) u | ⊗ | w (cid:105) ∧ | v (cid:105) = − a † v a u | w (cid:105) . Thus { a u , a † v } = 0.Now for each of the m pairwise nearly orthogonal planes, let {| u j (cid:105) , | v j (cid:105)} constitute an orthonormalbasis. Define X j = ( − a u j + a † u j )( a v j + a † v j ) Z j = 2 a v j a † v j − = a v j a † v j − a † v j a v j . (9)To understand this construction, observe that for orthonormal vectors | u (cid:105) , | v (cid:105) ∈ R n , and any | w (cid:105) ∈ Λ( R n ) with a u | w (cid:105) = a v | w (cid:105) = 0, the operators a u , a † u , a v , a † v fix the subspace spanned by | w (cid:105) , | v (cid:105) ∧ | w (cid:105) , | u (cid:105) ∧ | w (cid:105) , | u (cid:105) ∧ | v (cid:105) ∧ | w (cid:105) . In this basis, a u = (cid:18) (cid:19) a v = (cid:18) − 10 0 0 0 (cid:19) . Hence, ( − a u + a † u )( a v + a † v ) = (cid:18) (cid:19) a v a † v − = (cid:18) − − (cid:19) . The former matrix is σ X ⊗ σ X , and the latter matrix is I ⊗ σ Z , where σ X , σ Z are the standard Paulioperators. In particular, observe that X j = Z j = , X j Z j = − Z j X j .The above construction satisfies that if | u (cid:105) , | v (cid:105) , | u (cid:105) , | v (cid:105) are pairwise orthogonal, then [ X , X ] =[ X , Z ] = [ Z , X ] = [ Z , Z ] = 0. The reason we use two vectors to define each X j , Z j (instead ofjust taking X = a u + a † u , Z = 2 a u a † u − ) is to obtain the above commutation relationships. Since21 , Z are each quadratic in a u , a † u , a v , a † v , terms involving only a u , a † u , a v , a † v commute pastthem.Next, for nearly orthogonal planes we will show that the commutator norm (cid:107) [ S i , T j ] (cid:107) = O ( (cid:15) ),for i (cid:54) = j and S, T ∈ { X, Z } .If | u (cid:105) , | v (cid:105) are orthonormal, and | t (cid:105) = (cid:15) | u (cid:105) + √ − (cid:15) | v (cid:105) , then a t = (cid:15)a u + (cid:112) − (cid:15) a v = (cid:32) √ − (cid:15) (cid:15) 00 0 0 (cid:15) −√ − (cid:15) (cid:33) satisfies { a t , a u } = 0, { a t , a † u } = (cid:15) . In general, { a t , a u } = 0 { a t , a † u } = (cid:104) u | t (cid:105) . It follows that if |(cid:104) u | u (cid:105)| , |(cid:104) u | v (cid:105)| , |(cid:104) v | u (cid:105)| , |(cid:104) v | v (cid:105)| ≤ (cid:15) , then (cid:107) [ S , T ] (cid:107) = O ( (cid:15) ) for S, T ∈{ X, Z } . Indeed, X a u = ( − a u + a † u )( a v + a † v ) a u = − ( − a u + a † u ) (cid:2) a u ( a v + a † v ) − (cid:104) u | v (cid:105) (cid:3) = (cid:2) a u ( − a u + a † u ) − (cid:104) u | u (cid:105) (cid:3) ( a v + a † v ) + (cid:104) u | v (cid:105) ( − a u + a † u )= a u X − (cid:104) u | u (cid:105) ( a v + a † v ) + (cid:104) u | v (cid:105) ( − a u + a † u ) , implying (cid:107) [ X , a u ] (cid:107) ≤ |(cid:104) u | u (cid:105)| + |(cid:104) u | v (cid:105)| ≤ (cid:15) . Similarly, Z a u = (2 a u a † u − ) a u = 2 a u ( (cid:104) u | u (cid:105) − a u a † u ) − a u = a u Z + 2 |(cid:104) u | u (cid:105)| a u , implying (cid:107) [ Z , a u ] (cid:107) ≤ |(cid:104) u | u (cid:105)| ≤ (cid:15) . Thus (cid:107) [ S , T ] (cid:107) ≤ c (cid:15) for a fairly small constant cc