Unitary designs from statistical mechanics in random quantum circuits
UUnitary designs from statistical mechanicsin random quantum circuits
Nicholas Hunter-Jones
Perimeter Institute for Theoretical Physics,Waterloo, Ontario N2L 2Y5, Canada
Abstract
Random quantum circuits are proficient information scramblers and efficient generators ofrandomness, rapidly approximating moments of the unitary group. We study the convergenceof local random quantum circuits to unitary k -designs. Employing a statistical mechanicalmapping, we give an exact expression of the distance to forming an approximate design as alattice partition function. In the statistical mechanics model, the approach to randomness hasa simple interpretation in terms of domain walls extending through the circuit. We analyticallycompute the second moment, showing that random circuits acting on n qudits form approxi-mate 2-designs in O ( n ) depth, as is known. Furthermore, we argue that random circuits formapproximate unitary k -designs in O ( nk ) depth and are thus essentially optimal in both n and k .We can show this in the limit of large local dimension, but more generally rely on a conjectureabout the dominance of certain domain wall configurations. [email protected] a r X i v : . [ qu a n t - ph ] M a y ontents k -designs from statistical mechanics 12 k Random quantum circuits are invaluable constructions in both quantum information and quantummany-body physics. They are a rare example of a solvable model of strongly-coupled dynamicsand are efficient implementations of randomness. From a quantum information perspective, ran-dom quantum circuits are low-depth constructions of unitary k -designs [1, 2], rapid informationscramblers [3, 4], and essentially optimal decouplers [5]. Much has been understood about theirconvergence properties [1, 2, 6–10]. Moreover, recent benchmarks for demonstrating quantum ad-vantage involve the complexity of sampling the output distribution of random circuits [11]. Froma many-body viewpoint, quantum circuits are simple models of local chaotic dynamics and are avaluable resource in elucidating the onset of scrambling and thermalization in quantum systems.Recently, random circuits have been used to study both operator growth [12, 13] and the spreadingof entanglement [14–16] under chaotic evolution.In this work we study the convergence of local random quantum circuits to unitary k -designs,ensembles which emulate fully random unitaries by capturing the first k moments of the Haarmeasure on the unitary group, and thus yielding the power of random unitaries with much lowercomplexity. Designs are central in quantum information theory, with applications permeating abroad set of subfields. Prior work has shown that random circuits form approximate k -designs in acircuit depth that scales linearly in n and polynomially in k [2]. Our approach entails a synthesisof ideas from quantum information and condensed matter physics, utilizing an exact mapping tothe statistical mechanics of a lattice model. These techniques for random circuits were studiedin [12, 16], and we make extensive use of the ideas developed there.We focus on the frame potential, measuring the 2-norm distance to Haar-randomness, and showthat frame potential can be exactly written as the partition function of a spin system on a triangularlattice. For k = 2, the local degrees of freedom in the lattice model are simple and the approach toforming a 2-design can be understood in terms of decaying domain wall configurations. The secondframe potential is essentially exactly computable. We find that random circuits form (cid:15) -approximate2-designs in a depth n + log(1 /(cid:15) ) and precisely determine the coefficients.The statistical mapping is exact for general k , but the combinatorics of domain wall counting inthe lattice model for higher moments are less amenable to precise calculation. There is a single typeof non-intersecting domain wall in the second moment, but in higher moments we have complex1 Figure 1: The random circuits we consider are built from staggered layers of 2-site unitaries on n qudits of local dimension q , where each gate is drawn randomly from U ( q ).configurations of different types of interacting domain walls. We prove some general properties ofthe k -th partition function, which allows us to compute the leading order domain wall contribution.In the limit of large local dimension, we show that the k -design depth is O ( nk ). We can be precisein this limit, but for general local dimension a rigorous bound on the higher order terms thatcontribute to the partition function is not at hand. Nevertheless, given a conjecture that a simplesector of domain wall terms dominates the multidomain wall terms, which we provide evidence for,we argue that random quantum circuits form approximate k -designs in depth nk + k log k +log(1 /(cid:15) ).As the lower bound on the depth required to form a k -design is linear in n and k , this implies thatrandom circuits are essentially optimal implementations of unitary k -designs.We start by overviewing random quantum circuits and presenting definitions of the frame po-tential and the notion of an approximate k -design. In Sec. 2, we describe an exact mapping whichallows us to write the frame potential for random circuits as the partition function of a triangularlattice model. We then give a precise treatment of the second moment in Sec. 3, and understandthe non-zero configurations that contribute to the partition function in terms of domain walls inthe lattice model. In Sec. 4, we extend the discussion to general moments and argue that randomcircuits form k -designs in O ( nk ) depth. Random quantum circuits
The random quantum circuits (RQCs) we consider act on 1-dimensional chains of n qudits withlocal dimension q . We evolve in time by acting with staggered layers of 2-site random unitaries,acting on even links at even time steps and odd links at odd time steps, and where each 2-siteunitary is drawn Haar randomly from U ( q ). Explicitly, even time steps are given by the tensorproduct of 2-site unitaries (cid:78) i U i,i +1 for even i , and odd time steps by (cid:78) i U i,i +1 for odd i . Timeevolution to time t is given by acting with t layers of unitaries, such that the geometry of the circuitis fixed and not random, as shown in Fig. 1.Our goal is to compute the frame potential for random circuits, where the k -th frame potentialquantifies the 2-norm distance between between the k -fold channels. Focusing on the 2-norm willallow for exact calculation. The approach will be to give analytic expressions for the 2-norm andthen bound the approximate design depth from the diamond distance between the k -fold channels.We will restrict ourselves to considering moments less than the dimension k ≤ d = q n and notethat this is sufficient as O ( d ) gates is enough to implement a fully Haar random unitary.We first give a brief summary of the quantum information theoretic tools we need, including2he definition of an approximate k -design and the frame potential. A more detailed discussion withthe necessary definitions and proofs of some of the statements below is given in App. A. Approximate k -designs The Haar measure is the unique left/right invariant measure on the unitary group U ( d ). We areinterested in when a set of unitaries captures moments of the full unitary group. Consider anensemble of unitaries E , a subset of U ( d ) equipped with a probability measure. For an operator O acting on the k -fold Hilbert space H ⊗ k , the k -fold channel with respect to E is defined asΦ ( k ) E ( O ) = (cid:90) E dU U ⊗ k ( O ) U †⊗ k . (1)In this sense we ask when the average of an operator over the ensemble E equals an average overthe full unitary group. A unitary k -design is an ensemble E for which the k -fold channels are equalfor all operators O , Φ ( k ) E ( O ) = Φ ( k )Haar ( O ) . (2)Forming a k -design means we exactly capture the first k moments of the Haar measure, andreproducing higher and higher moments corresponds to better approximating the full unitary group.For k = 1, any basis for the algebra of operators of H , including the Pauli group, is a 1-design. Butfor general k , very few constructions of exact k -designs are known, with the exception of k = 2 and k = 3 [17–21]. Instead, one may relax this notion and ask when an ensemble of unitaries is close to forming a k -design. Definition (approximate k -design) . For (cid:15) > , an ensemble of unitaries E is an (cid:15) -approximate k -design if the diamond norm of the difference of k -fold channels is (cid:13)(cid:13) Φ ( k ) E − Φ ( k )Haar (cid:13)(cid:13) (cid:5) ≤ (cid:15) . (3)There are varying definitions of approximate designs, some of which involve different norms, butthe definitions are equivalent up to powers of the dimension; see [22] for a nice overview. Lastly, wenote that [2] used a slightly stronger definition in terms of the complete positivity of the differencein superoperators, but this will only affect our bounds by an additional factor of the dimension. Frame potential
The frame potential for an ensemble of unitaries E was first discussed in the context of unitarydesigns in [23, 24]. More recent interest involves the frame potential as a diagnostic of chaotic dy-namics [25–28]. Specifically, [25] related the frame potential to both averaged out-of-time-orderedcorrelators and measures of complexity, and [26] computed the frame potentials for random Hamil-tonian evolution. Definition (frame potential) . The k -th frame potential of an ensemble of unitaries E is defined asa double average over the ensemble F ( k ) E = (cid:90) U,V ∈E dU dV (cid:12)(cid:12) Tr( U † V ) | k . (4)The frame potential for any ensemble is lower bounded by its Haar value as F ( k ) E ≥ F ( k )Haar , (5)3ith equality if and only if the ensemble forms a k -design. The Haar value of the frame potentialis F ( k )Haar = k ! for moments k ≤ d .Relating this measure of Haar-randomness to our definition of an approximate design, thedifference in the k -th frame potentials of an ensemble E and the Haar ensemble bounds the diamondnorm of the difference in k -fold channels as (cid:13)(cid:13) Φ ( k ) E − Φ ( k )Haar (cid:13)(cid:13) (cid:5) ≤ d k (cid:0) F ( k ) E − F ( k )Haar (cid:1) . (6)We present the proofs of both of the above statements in Eq. (5) and Eq. (6) in App. A. Previous results
Harrow and Low [1] (with [29]) studied local random quantum circuits with randomly applied gates,and by looking at the mixing time of the Markov chain on Pauli strings, showed that they formapproximate 2-designs in O ( n ) depth. In their random circuits each time step corresponds to theapplication of a single gate, whereas in our parallelized random circuits each time step consists of O ( n ) gates, and thus the time scales differ by a factor of n .These results were extended in [10] by studying the gap of the moment operator, showing thatrandom circuits form 3-designs. The gap of the second moment operator was also computed exactlyin [8], and properties of the k -th moment operator were studied in [9], both for 2-local all-to-allcoupled circuits.Brand˜ao, Harrow, and Horodecki [2, 30] further extended this approach to general k , showingthat random circuits form approximate unitary k -designs in a depth O ( nk ). More recently,[31] studied higher dimensional random circuits, demonstrating that they form k -designs in depth O ( n /D poly( k )), with the degree of the polynomial depending on the dimension D of the lattice.Furthermore, it was shown in [2] that the lower bound on the depth required for a 1D randomcircuit to form a k -design is linear in n and k , so the dependence on k cannot be improved bymore than polynomial factors. Progress towards optimality in k was made in [32] for randomHamiltonian evolution, where they found convergence to k -designs in O ( n k ) steps, for momentsup to k = o ( √ n ). It has also been shown [33] that Brownian quantum circuits [34] form designsin O ( nk ) depth, with the same dependence on k as in [2]. Lastly, we note that [35] numericallyinvestigated the gap of the moment operator and found evidence for a linear growth in design fromthe first few moments. Our results
We show that the frame potential for random quantum circuits on n qudits of local dimension q and circuit depth t can be exactly written as a partition function of a triangular lattice model F ( k )RQC = (cid:88) { σ } (7)where the local degrees of freedom are permutations σ ∈ S k , and the lattice is of width n g = (cid:98) n/ (cid:99) ,depth 2( t − k = 2 frame potential is essentially exactly computable and thenonzero spin configurations have a simple interpretation in terms of domain walls separating regions4f local identity and swap permutations. By counting the domain wall configurations we computethe depth at which random circuits form (cid:15) -approximate unitary 2-designs to be t ≥ C (cid:0) n log q + log n + log 1 /(cid:15) (cid:1) with C = (cid:18) log q + 12 q (cid:19) − , (8)with the linear term ≈ . n for q = 2 and asymptoting to 2 n for q → ∞ .The partition function for general k receives contributions from more complicated domain wallconfigurations, where each domain wall represents an element of the generating set of transpositionsfor the symmetric group S k . We prove some general properties for the plaquette terms constitutingthe k -th partition function and show that only a simple sector of domain wall configurations con-tributes at leading order in 1 /q , which allows us to compute the depth at which we form a k -designin the large q limit to be t k = O ( nk ) . (9)Furthermore, we argue that, given a conjecture that the sector of single domain wall configurationsbounds the multi-domain wall configurations at any finite q , random quantum circuits are optimalgenerators of randomness achieving the lower bound on the design depth. Random unitary circuits admit an exact mapping to a classical spin system, allowing for a simpli-fication in computation and analytic treatment of the moments. A classical mapping for randomtensor networks was first discussed in [36], but explicit details for the circuits we consider here weredescribed in [12, 16]. Specifically, [12] used the statistical mechanical mapping to exactly computethe out-of-time ordered correlation function in random circuits, and [16] extended the techniquesfor a replica calculation of the R´enyi entropies.The quantity of interest here is the k -th frame potential for random unitary circuits, defined as F ( k )RQC = (cid:90) U,V ∈ RQC dU dV (cid:12)(cid:12)
Tr( U † t V t ) | k , (10)for RQCs evolved to time t . Visualizing this as a circuit diagram, for a single U † V we have (11)with the U time increasing left to right on the blue circuit and the V time increasing right to left onthe purple circuit. The trace enforces periodic boundary conditions in time in the circuit. For the k -th frame potential we have k copies of U and V and their complex conjugates. First note thatwe can combine a single layer on each side of the U and V circuits using the left/right-invarianceof the Haar measure, i.e. in the middle and through the trace. As we are averaging over each 2-site5nitary in the U and V circuits independently, the frame potential is equivalently an average ofmoments of traces of depth 2( t −
1) circuits F ( k )RQC = (cid:90) RQC dU (cid:12)(cid:12) Tr( U t − ) | k . (12)We want to consider k -th moments of the random circuit and compute averages over all 2-siteunitaries. If we imagine stacking the k copies of the circuit and its conjugate on top of each other,then we see that performing an average over each 2-site unitary of the form U ⊗ k ⊗ U †⊗ k will giveindex contractions with neighboring gates on the k -fold space.We review Haar integration in more detail in App. A, but briefly recall that averages overpolynomials of random unitaries take the form (cid:90) dU U ⊗ k(cid:126)ı,(cid:126) ⊗ U †⊗ k(cid:126)(cid:96), (cid:126)m = (cid:88) σ,τ ∈ S k δ σ ( (cid:126)ı | (cid:126)m ) δ τ ( (cid:126) | (cid:126)(cid:96) ) W g ( σ − τ, d ) , (13)where the i ’s and j ’s are the indices of the U ’s, and (cid:96) ’s and m ’s are the indices of the U † ’s.Heuristically, Haar integration is a prescription for index contraction, telling us to contract in theingoing indices of U to the outgoing indices of U † and the outgoing indices of U to the ingoingindices of U † , both indexed by a permutation with a weight that is a function of the permutations.Explicitly δ σ ( (cid:126)ı | (cid:126) ) = δ i ,j σ (1) . . . δ i k ,j σ ( k ) and W g ( σ, d ) is the Weingarten function on elements of thesymmetric group S k .Thinking about each gate stacked on top of the 2 k copies of itself, and denoting the gates as U ij and U † (cid:96)m , the Weingarten formula then tells us to contract i → m and j → (cid:96) indices, each withrespect to an element of S k . We can represent this with an effective vertex [12]:The blue gates are U ’s and the yellow gates are U ∗ ’s (i.e. from folding the circuit of U † ’s). Wehave incoming im indices and outgoing j(cid:96) indices, and by averaging the gate we generate indexcontractions in each of the legs of the vertex with respect to a permutation at the node.The red nodes are permutations σ ∈ S k , the blue nodes are τ ∈ S k , the horizontal lines areweights given by the Weingarten function W g ( σ − τ, q ), and the diagonal lines denote the indexcontractions between the legs of two gates in the circuit. These contractions between legs give aweight determined by an inner product between permutations: (cid:104) σ | τ (cid:105) = q (cid:96) ( σ − τ ) , where (cid:96) ( σ − τ )is the length of the cycle-type of the permutation product, i.e. the number of closed loops in theproduct. For k = 2 we have (cid:104) σ | τ (cid:105) = σ τ = q (cid:96) ( σ − τ ) , (14)where the four lines are the contractions between gates in the two copies of the circuit and itsconjugate, and each permutation gives contraction between the two top lines and the two bottomlines, giving factors of the local dimension q . For instance, the inner products we find for k = 2are: (cid:104) I | I (cid:105) = q , (cid:104) S | S (cid:105) = q , (cid:104) I | S (cid:105) , and (cid:104) S | I (cid:105) = q .Integrating over each 2-site unitary in the circuit means we replace each gate with the effectivevertex above and sum over all σ, τ ∈ S k . The result is that the k -th frame potential can be written as6igure 2: The k -th frame potential can be written as the partition function of a spin system ona hexagonal lattice with local S k spins (left). By summing over the blue nodes we can defineeffective plaquette terms and write the frame potential as the partition function on a triangularlattice (right). In both figures time runs from left to right and periodic boundary conditions intime means the red nodes on the ends of the circuit are identified.the partition function of a hexagonal lattice model on the left in Fig. 2, summing over permutationsin S k at each node and assigning a weight as described above. We have a depth 2( t −
1) latticewith periodic boundary conditions in time, so the leftmost and rightmost red nodes in the latticeare identified.The k -th frame potential is exactly equal to a sum over spin configurations on a hexagonallattice. A priori, this reduction to an spin system is not too enlightening. We must evaluate allspin configurations, many of which are negative, and carefully assign weights to a given configurationas prescribed above. But, as was described in [12], some substantial simplifications occur when wesum over certain local spins.Performing the sum over the blue nodes in the partition function, the local τ spins, we can thendefine effective plaquette terms, which are functions of three permutationswhere J σ σ σ ≡ (cid:88) τ ∈ S k σ σ σ τ (15)thereby reducing frame potential to a partition function on a triangular lattice as shown in Fig. 2. With this reduction, the frame potential is now equal to the partition function of local S k spinson a triangular lattice F ( k )RQC = (cid:88) { σ } (cid:89) (cid:47) J σ σ σ = (cid:88) { σ } (16) We will refer to the local degrees of freedom in the lattice model as spins, keeping in mind they are elements ofthe symmetric group S k . Had we not used the invariance of U ( q ) to absorb a layer of gates between U and V † in Eq. (11), we would havehad a boundary vertex between the two circuits, where summing over the τ spins on the blue node gives a δ -functionfor σ spins → = δ σ σ . n g = (cid:98) n/ (cid:99) sites across, with n g denoting the number of gates in a layer, and2( t −
1) sites deep, with periodic boundary conditions in time. The plaquettes are functions ofthree permutations, written explicitly in terms of the Weingarten functions and permutation innerproducts as J σ σ σ = σ σ σ = (cid:88) τ ∈ S k W g ( σ − τ, d ) q (cid:96) ( τ − σ ) q (cid:96) ( τ − σ ) . (17)We stress that the lattice partition function is not physical, but is simply a way to analyticallyexpress moments of the random circuit. Furthermore, we note that the random circuit quantitiesone wishes to compute are entirely encoded in the boundary conditions of the lattice model, whetherthey be correlation functions [12] or R´enyi entropies [16]. In this sense, the frame potential is simpleas it merely enforces periodic boundary conditions in the lattice model. k = 2 plaquette terms We reduced the computation of the frame potential to a combinatorial problem of enumeratinglattice configurations of spins. In the k = 2 partition function the local spins are σ ∈ S = { I , S } ,the identity and swap permutations. Using Eq. (17), we can compute the possible plaquette termsfor the partition function of the second moment [12]: I II = 1 , I SS = 0 , I I S = I S I = q ( q + 1) ,S SS = 1 , S II = 0 , S S I = S I S = q ( q + 1) . (18)Plaquette terms in the first column mean that lattice configurations of all I ’s or all S ’s haveweight one. Moreover, the plaquette terms in the second column ensure that any spin configurationwith a single permutation differing from the rest will have weight zero. We can understand thepossible contributions to the partition function by thinking about domain walls separating regionsof identity and swap permutations. The nonzero plaquette terms in Eq. (18) mean that, propa-gating through the network in the time direction, a domain wall can either move left or right at aplaquette. Given the periodic boundary conditions in time, the non-zero subleading contributionsin the partition function arise from domain walls between I ’s and S ’s running through the circuit F (2)RQC = 2 + (cid:88) domain wall configurations , (19)where each domain wall contributes a factor of q/ ( q + 1) for each time step. We see that notonly must we sum over single domain walls, but also configurations of multiple domain walls, asshown in Fig. 3. Moreover, Eq. (18) guarantees that domain walls cannot end, and thus, giventhe periodic boundary conditions, no closed loops are allowed. The same reasoning holds for theboundary plaquettes; domain walls may enter through the boundary, but cannot exit. Periodicboundary conditions ensure that all such configurations are zero.8igure 3: Examples of nonzero contributions to the k = 2 partition function: a single and doubledomain wall configuration, where the domain walls separate regions of local identity and swappermutations. k -designs from the ground states In this picture it is clear where k !, the minimal Haar-random value of the frame potential, comesfrom. The plaquette terms have weight one when evaluated on the same spin σ ∈ S k . We discussthis in Sec. 4 and prove this for general k in App. B. Therefore, in the k -th frame potential thereare k ! configurations with unit weight. In the language of the spin partition function, there isan energy cost associated to having different neighboring spins. The ground states of the latticemodel are the k ! spin configurations with all sites are equal. The excited states correspond todomains of differing spins, where there is an energy cost in 1 /q for differing neighboring spins. Wewill give a more extensive discussion of the k -th frame potential in Sec. 4, where we compute thecontributions from excited states, but in the strict limit of large local dimension q → ∞ , we seethat F ( k )RQC ≈ k ! + O (1 /q ). Some exact results for RQCs
We briefly comment on some straightforward exact results. The frame potential for RQCs at time t = 0 is simply the contribution from the identity F ( k )RQC ( t = 0) = d k = q nk . At time t = 1, withone layer of circuit evolution, we have a tensor product of 2-site unitaries. The frame potential isa product of the moments of traces of the gates F ( k )RQC ( t = 1) = ( k !) n/ , for moments k ≤ q andassuming an even number of sites. For k > q , the moments of traces of the 2-site unitaries areknown combinatorial factors.The first moment of random circuits is also simple. Every gate we Haar average generates asingle index contraction with its neighboring gates, factors of the dimension exactly canceled bythe Weingarten coefficients. In the language of our lattice model, there is only one local degree offreedom in S and only one kind of plaquette term with weight 1. The k = 1 frame potential is F (1)RQC ( t ) = 1 , (20)and thus random quantum circuits form exact 1-designs for all q , n , and t . Computing the k -th frame potential exactly from the statistical mechanics model might be difficult,but for k = 2, where we have local Ising-like spins and simple domain walls, an analytic treatmentis more straightforward. As we showed, the k = 2 frame potential for random quantum circuits issimply the partition function for S spins on a directed triangular plaquette model. We computed9he exact plaquette terms in Eq. (18) and argued that the k = 2 partition function is a sum overall domain wall configurations on the triangular lattice F (2)RQC = 2 (cid:18) (cid:88) wt ( q, t ) + (cid:88) wt ( q, t ) + . . . (cid:19) . (21)Two such possible configurations are shown in Fig. 3. We can upper bound this quantity directly.There is a spin flip symmetry of the partition function, so an overall factor of 2 in the sum.The domain wall configurations must start and end at the same point due to periodic boundaryconditions in time. Each domain wall contributes a weight qq +1 per time step of the depth 2( t − F (2)RQC = 2 (cid:18) c ( n, t ) (cid:18) qq + 1 (cid:19) t − + c ( n, t ) (cid:18) qq + 1 (cid:19) t − + . . . (cid:19) , (22)where c i is the number of spin configurations with i domain walls. For a single domain wall, thereare at most 2( t −
1) choose t − n g −
1) starting points, where n g = (cid:98) n/ (cid:99) is the number of gates. Iteratingup to configurations of ( n g −
1) domain walls, we may bound the frame potential as F (2)RQC ≤ (cid:18) n g − (cid:18) t − t − (cid:19) (cid:18) qq + 1 (cid:19) t − + (cid:18) n g − (cid:19)(cid:18) t − t − (cid:19) (cid:18) qq + 1 (cid:19) t − + . . . + (cid:18) n g − n g − (cid:19)(cid:18) t − t − (cid:19) n g − (cid:18) qq + 1 (cid:19) n g − t − (cid:19) , (23)with combinatorial factors enumerating multiple domain wall configurations and their associatedweight. Continuing, we sum the binomial expansion in n g F (2)RQC ≤ (cid:18) (cid:18) t − t − (cid:19) (cid:18) qq + 1 (cid:19) t − (cid:19) n g − < (cid:18) (cid:18) qq + 1 (cid:19) t − (cid:19) n g − , (24)upper bounding the binomial. We see that the second frame potential for random circuits decaysas F (2)RQC ≈ /q t ) n , reaching its Haar value at a time scale t ∼ log n .Note that the bound is overcounting the domain wall configurations in two ways: we are con-sidering a finite number of sites and the domain walls cannot cross spatial boundaries, and domainwalls cannot cross, so the number of the paths that multiple domain walls can take are restricted.Lastly, we note that the function of q in Eq. (24) is exponentially decaying in time for all localdimensions q , i.e. q = 2 and greater. RQC 2-design time
We now compute the 2-design time for random quantum circuits. The difference in frame potentialsbounds the diamond distance between the channels as (cid:13)(cid:13) Φ (2)RQC − Φ (2)Haar (cid:13)(cid:13) (cid:5) ≤ d (cid:0) F (2)RQC − F (2)Haar (cid:1) . (25)10sing the upper bound in Eq. (24), we can compute depth at which random circuits form an (cid:15) -approximate 2-design from q n (cid:32) (cid:18) (cid:18) qq + 1 (cid:19) t − (cid:19) n g − − (cid:33) / ≤ (cid:15) . (26)Taking the log of both sides, we expand out the binomial in the log and upper bound each termin terms of the leading order term, i.e. the multidomain wall contributions are all bounded by thesingle domain wall term. After some manipulation, we find the circuit depth at which we form an (cid:15) -approximate 2-design to be t ≥ C (cid:0) n log q + log n + log 1 /(cid:15) (cid:1) with C = (cid:18) log q + 12 q (cid:19) − . (27)To get a sense of the scaling of n with q , we see that for qubits with local dimension q = 2 thelinear term is t ∼ . n , and in the limit of large local dimension q → ∞ we find t ∼ n .Our analytic treatment of the second moment gives the 2-design time of t = O ( n + log 1 /(cid:15) ),reproducing the known result in [1, 2]. Given that the frame potential reaches its minimal value inlog n time, we see from Eq. (27) that it takes an additional O ( n ) time to get close in norm. Counting domain walls
Above we opted for a simple bound on the allowed domain wall configuration, but it is possibleto compute the combinatorial coefficients exactly, accounting for the presence of boundaries andinteractions between domain walls. The problem is equivalent to counting the number of non-intersection random walks on a finite 1D lattice. Consider two non-intersecting domain walls andfor the moment ignore the effects of the boundaries. We can use the method of images [37] tocompute the number of non-intersecting configurations. For two random walkers starting at points x and y , respectively, and ending at the same points after 2( t −
1) time steps, there are at most (cid:0) t − t − (cid:1) possible paths. To account for the crossings, note that any intersection of the two randomwalks from x → x (cid:48) and y → y (cid:48) can also be thought of as a path from x → y (cid:48) and y → x (cid:48) . So thenumber of possible paths for two non-intersecting random walkers to return to their starting pointsafter 2( t −
1) time steps is (cid:18) t − t − (cid:19) − (cid:18) t − t − − | x − y | (cid:19) , (28)where | x − y | is the distance between the two walkers. Similarly, we can accommodate the effectsof a boundary by again using a reflection. For a single random walker x sites from the boundaryreturning to site x after 2( t −
1) steps, any path hitting the boundary can be thought of as a randomwalk from − x to x . Thus the number of single domain wall configurations, accounting for bothboundaries and summing over all starting points, is c ( n, t ) = n g − (cid:88) x =1 (cid:18)(cid:18) t − t − (cid:19) − (cid:18) t − t − − x (cid:19) − (cid:18) t − t − − n g − x ) (cid:19)(cid:19) . (29)But this only takes into account paths crossing one of the boundaries any number of times. Forvery long times, we need to account for paths that hit the left boundary and then the rightboundary, etc. We can again count the paths by repeating the method of images for multipleboundary intersections. Furthermore, we can iterate the same procedure to count multidomain11all configurations. For p random walkers, one may consider a single random walk moving in p dimensions, with a restriction on the coordinates from the non-intersection of the walkers. Iteratingthe method of images, one can determine the overall distribution of multiple random walks [37–39].This will give an analytic expression for the multidomain wall contributions in the presence ofboundaries. Periodic boundary conditions
If we instead start with a 1D array of qudits with periodic boundary conditions on the chain, itis straightforward to see that the above bound in Eq. (24) still holds: even if paths can wraparound the circuit, there are still at most 2 t − paths for a single domain wall after 2( t −
1) timesteps. Moreover, periodic spatial boundary conditions would mean that configurations with an oddnumber of domain walls would be disallowed. The leading order contribution would arise fromdouble domain wall configurations and thus the frame potential would still decay in log n time,albeit slightly quicker than with open boundary conditions. k -designs from statistical mechanics We now proceed to discussing higher moments of random circuits. The statistical mapping discussedin Sec. 2 allows us to write the k -th frame potential for random quantum circuits as a classicalpartition function of spins σ ∈ S k on a triangular lattice of width n g and depth 2( t −
1) with periodicboundary conditions in time. In the second moment, the exited states were spin configurations withone type of domain wall separating regions of identity and swap permutations. In this sense thedomain wall represented a swap, passing through the domain wall took us from I spins to S spins.Moreover, the domain walls could not intersect, pair create, or annihilate, and their counting wasamenable to a simple combinatorial treatment.At higher k , this simple picture no longer holds. We have multiple different types of domainwalls which can interact nontrivially, including interactions where domain walls end. Recall that theplaquette terms J σ σ σ are functions of three permutations σ ∈ S k , defined as a sum of Weingartenfunctions and permutation inner products over an internal permutation as J σ σ σ = (cid:88) τ ∈ S k W g ( σ − τ, q ) q (cid:96) ( σ − τ ) q (cid:96) ( σ − τ ) . (30)Computing these quantities for all σ ∈ S k will tell us the allowed terms in the k -th partitionfunction. The types of plaquette terms can be understood in terms of domain walls between thedifferent permutations. As in [16], domain walls for the k -th moment will denote transpositionson k elements, a generating set for the symmetric group S k . For instance, in the third moment,if we have neighboring spins { , , } and { , , } , we say they are separated by a domain wallrepresenting the (23) transposition. In general, there are (cid:0) k (cid:1) transpositions and any two elementsof S k differ by at most k − k = 3, we find additional complications asthere is an interaction term with 4 incoming and 2 outgoing domain walls. We prove some generalproperties of domain walls in App. B and give the explicit weights of plaquette terms for the firstfew moments. 12 Figure 4: An example of a domain wall annihilating at a vertex, creating a closed loop in the circuit.The domain walls represent transpositions, generators of S k , which we denoted with different colors. Single domain wall sector
For general k , there are significant complications in the multi-domain wall contributions to thepartition function. But as we will see, the single domain wall sector remains simple. This stemsfrom two facts about the plaquette terms for general k , which we prove in App. B. The first is thatfor spins σ ∈ S k , we have J σσσ = 1 when all spins are the same, and that J σσ (cid:48) σ (cid:48) = 0 for σ (cid:54) = σ (cid:48) , i.e.the weight for no domain walls is just one and interactions with just two domain walls annhilatingare disallowed. This means we cannot have closed loops in the circuit. These statements both arisefrom the fact that contracting a U with a U † in the average gives the identity.The other fact is that we cannot annihilate incoming domain walls in plaquettes with only oneoutgoing domain wall. This arises from the statement that if σ and σ in J σ σ σ only differ by asingle transposition, then the resulting Haar integral reduces to a second moment calculation, whichcan only give an identity or swap. This translates to only allowing one incoming domain wall ifthere is only one outgoing domain wall, and forbids domain wall annihilation in the single domainwall sector. More generally, incoming domain walls can annhiliate in plaquettes with multipleoutgoing domain walls, but these properties guarantee the independence of the single domain wallsector.We now count the configurations of single domain walls. A domain wall corresponds to a singletransposition, of which there are (cid:0) k (cid:1) for spins σ ∈ S k . Accounting for the k ! spin flip symmetryof the lattice, we then count the directed random walks through the depth 2( t −
1) lattice, with( n g −
1) starting points for a single domain wall and assigning the weight, we find (cid:88) wt ( q, t ) ≤ ( n g − (cid:18) k (cid:19)(cid:18) t − t − (cid:19)(cid:16) qq + 1 (cid:17) t − . (31) Multi-domain wall sector
For the k -th moment of the random circuit, we will have contributions from domain walls which caninteract and can pair create and annihilate. For instance, at k = 3 we already have an interactionvertex with 4 ingoing domain walls and 2 outgoing, which permits spin configurations with closedloops, as shown in Fig. 4. This means we cannot separate the contributions in the partition functioninto sectors of differing numbers of domain walls and simply count them.We gave a bound on single domain wall configurations above. Similarly, for double domain wallconfigurations with no intersections, we have the bound < ( n g − (cid:18) k (cid:19) (cid:18) t − t − (cid:19) (cid:16) qq + 1 (cid:17) t − . (32)13here are at most 2( n g − k −
1) domain walls at t = 0, as any permutation in S k can be reachedwith at most k − ∼ nk multidomain wall configurations with no intersectionsbetween domain walls, all can be bounded by powers of the single domain wall contribution.The technical hurdle that arises for higher moments is in dealing with possible intersections.Interaction terms between multiple different types of domain walls are allowed with weights de-pending on the types of incoming domain walls. Some of the interaction terms up to k = 4 aregiven in App. B, but we have computed all such terms up to k = 6. Even for two domain walls witha single intersection, we may then have many configurations of multiple closed loops in the lattice.To rigorously bound their contribution to the partition function we must bound the number of allsuch possible configurations. Nevertheless, these interactions are suppressed by many factors of thelocal dimension, seemingly ensuring the dominance of the single domain wall sector. RQC k -design time Given what we have understood so far, we can sketch how the k -design time should arise fromthe k -th frame potential. We have the contribution from the ground states and single domain wallsector, plus higher order contributions F ( k )RQC = k ! (cid:18) n g − (cid:18) k (cid:19)(cid:18) t − t − (cid:19)(cid:16) qq + 1 (cid:17) t − + . . . (cid:19) . (33)Recalling how the frame potentials bound the distance between k -fold channels, (cid:13)(cid:13) Φ ( k )RQC − Φ ( k )Haar (cid:13)(cid:13) (cid:5) ≤ d k (cid:0) F ( k )RQC − F ( k )Haar (cid:1) , (34)we see that the single domain wall sector will give the depth at which we form an approximate k -design to be t k ∼ nk log q + k log k + log(1 /(cid:15) ), up to constant factors. For moments k ≤ q n , wefind a depth O ( nk ). k -designs at large q The analysis for general moments becomes simple in the limit of large local dimension q . We provein App. B that the only plaquette terms that contribute at order 1 /q are single domain walls. Moregenerally, interaction terms with l outgoing domain walls will contribute at most at order 1 /q l ,with additional penalties by factors of 1 /q if we have domain walls annihilating in an interactionterm. So more complicated interactions are generally greatly suppressed in q , with only single freelypropagating domain walls contributing at leading order in 1 /q . Thus in the q → ∞ limit we canbound the higher order domain wall terms in Eq. (33) and find the difference in the k -th framepotentials to be F ( k )RQC − F ( k )Haar ≤ k ! (cid:18) nk (cid:18) q (cid:19) t − + O (cid:18) q t (cid:19)(cid:19) , (35)from which we recover the k -design time in the large q limit t k ≥ C (cid:0) nk log q + k log k + log( nk ) + log(1 /(cid:15) ) (cid:1) , (36)where C = log − ( q/ -designs at finite q For a given k , we can explicitly compute the plaquette terms as functions of q , but the number ofunique interaction terms grows rapidly with k . We can understand their behavior to leading orderin 1 /q , but it is unclear how much the exact functions matter in trying to bound the higher ordercontributions to the k -th partition function, or if additional subtleties arise for exponentially largemoments and small q .One issue that arises when considering higher moments is that the Weingarten functions havepoles in q at {± , . . . , ± ( k − } . Thus, one might worry about the partition function for k > q ,there are a few reasons this is not an issue, which we discuss in more detail in App. B. For momentsgreater than the dimension, the Weingarten function is modified (restricting the sum over integerpartitions) to free the denominator of zeroes. But we can drop this restriction and will find that thepoles in the plaquette terms will cancel for any physical value of the dimension q ∈ N . Moreover,the plaquette rules for general k we derive in App. B arise from properties of permutations actingon the k -fold space and hold for any q .We end this section on a slightly more speculative note. Taking the limit of large local dimensionensures the dominance of the single domain wall terms, but such a limit is likely not necessary.Configurations with multiple domain walls decay rapidly, and the more complicated the types ofinteractions, the stronger the suppression. Furthermore, the interaction terms with annihilatingdomain walls have negative weight, further reducing their contribution to the partition function.Thus, we conjecture that the single domain wall terms dominate for general k . Conjecture.
The single domain wall sector of the lattice partition function dominates the mul-tidomain wall sectors for higher moments k and any local dimension q . Lastly, a lower bound on the depth is known (Prop. 8 in [2]). For (cid:15) ≤ / k ≤ d / , an (cid:15) -approximate k -design on n qudits of local dimension q must have circuit depth at least t k ≥ nk q log( nk ) . (37)If the single domain wall contribution individually bounds each of the ∼ nk domain wall terms,then the depth at which random circuits form an approximate k -design is O ( nk ), with a dependenceon n and k that is essentially optimal. We studied the convergence of random quantum circuits to unitary k -designs and by exactly ex-pressing the frame potential as a lattice partition function, were able to give an analytic treatmentof the second moment, showing that random circuits form approximate 2-designs in O ( n ) depth.By then proving some properties of the partition function for the k -th moment, we then showedthat in large q limit random circuits form k -designs in O ( nk ) depth. We further conjectured thatthe terms corresponding to single domain walls in the spin system bound the multi-domain wallcontributions for any local dimension q , from which it holds that the k -design time is O ( nk ) andthus random circuits are optimal implementations of randomness. A rigorous bound on the higherorder terms in the k -th partition function is needed to make this precise, and we hope to return tothis in future work.As we have seen, the statistical mechanical mapping renders certain random circuit calculationsanalytically tractable. It would be interesting to see if these techniques could be applied to the15igher-dimensional random circuits considered in [31]. Moreover, we focused on circuits with afixed geometry, which gave a partition function on a regular triangular lattice. Considering randomcircuits with gates applied randomly to neighboring pairs of qudits should give the partition functionof a spin system on random triangular tilings of the strip. Furthermore, there has been recentinterest in other random circuit models with conservation laws [40, 41], local symmetries includingtime-reversal symmetry [42], and Floquet random circuits [43, 44]. It would be interesting to see iflattice models could be defined for each of these circuits. For instance, one could define a latticemapping for the circuits considered in [42] using Weingarten calculus for the different symmetrygroups and show that circuits built from random orthogonal gates converge to the Haar measureon O ( d ). It would also be interesting to investigate whether the charge conserving random circuitsin [40, 41] form k -designs within fixed charge sectors. Acknowledgments
The author would like to thank Fernando Brand˜ao, Jacob Bridgeman, Richard Kueng, Zi-Wen Liu,Saeed Mehraban, Beni Yoshida, and especially Adam Nahum for helpful discussions, as well as´Alvaro Alhambra and Thom Bohdanowicz for discussions and comments on the draft. The authoralso thanks the IQIM at Caltech, McGill University, and UC Berkeley for hospitality during thecompletion of part of this work. Research at Perimeter Institute is supported by the Governmentof Canada through the Department of Innovation, Science and Economic Development Canada andby the Province of Ontario through the Ministry of Research, Innovation and Science.
A Approximate designs and random unitaries
In this appendix we quickly review some definitions related to approximate k -designs and Haarintegration. We introduced the notion of an approximate k -design in Def. 1 as the distance of the k -fold channels in diamond norm. After providing a few relevant definitions we will also show howto bound the diamond norm in terms of the frame potential. k -fold channels The k -fold channel of an operator O with respect to an ensemble E of unitaries is defined asΦ ( k ) E ( O ) ≡ (cid:90) dU U ⊗ k ( O ) U †⊗ k , (38)here written for a continuous ensemble. As we discussed, an exact unitary k -design is an ensemble E ,a subset of the unitary group equipped with some probability measure, for which the k -fold channelsare equal Φ ( k ) E ( O ) = Φ ( k )Haar ( O ), for all operators acting on the k -fold Hilbert space O ∈ A ( H ⊗ k ).It will also be convenient to introduce the moment operator for an ensemble E , closely relatedto the k -fold channel. The k -th moment operator (cid:98) Φ ( k ) E is defined as (cid:98) Φ ( k ) E ≡ (cid:90) dU U ⊗ k ⊗ U †⊗ k . (39)The k -th moment operators for an ensemble E and the Haar ensemble are also equal (cid:98) Φ ( k ) E = (cid:98) Φ ( k )Haar if and only if E forms a k -design. 16 perator norms The Schatten p -norm of an operator O is defined for p ≥ (cid:107)O(cid:107) p ≡ (cid:0) Tr |O| p (cid:1) /p , which obeysmonotonicity of norms (cid:107)O(cid:107) q ≤ (cid:107)O(cid:107) p for q ≥ p . The superoperator norm of a quantum channel isdefined for any p ≥ (cid:107) Φ (cid:107) p → p ≡ sup O(cid:54) =0 (cid:107) Φ( O ) (cid:107) p (cid:107)O(cid:107) p . (40)The diamond norm of a quantum channel Φ is defined as (cid:107) Φ (cid:107) (cid:5) ≡ sup d (cid:107) Φ ⊗ Id d (cid:107) → , (41)where Id d is the identity channel on a d -dimensional ancilla. The diamond norm captures thedistinguishability of quantum channels. One relation between the diamond norm and the operator2-norm we will need is (cid:107) Φ (cid:107) (cid:5) ≤ d k (cid:107) Φ (cid:107) → , for Φ acting on k -fold operators. Approximate k -design We define an (cid:15) -approximate k -design as an ensemble of unitaries E such that the k -fold channelwith respect to E is (cid:13)(cid:13) Φ ( k ) E − Φ ( k )Haar (cid:13)(cid:13) (cid:5) ≤ (cid:15) , (42)meaning that the ensembles are close with respect to the diamond norm. The notion of an ap-proximate design makes precise a distance to randomness, capturing how close an ensemble is toreplicating moments of U ( d ). There are other definitions of approximate designs, some employingdifferent norms, which bound each other up to factors of the dimension, as is reviewed in [22]. Frame potential
In general, diamond norms are difficult to compute. There are other quantities which capture thedistance of an ensemble to the Haar ensemble which are more tractable. The frame potential isdefined as a double average over an ensemble of unitaries E as [23, 24] F ( k ) E = (cid:90) U,V ∈E dU dV (cid:12)(cid:12) Tr( U † V ) (cid:12)(cid:12) k . (43)The frame potential is related to the 2-norm distance of the ensembles and is lower bounded by itsHaar value F ( k ) E ≥ F ( k )Haar where F ( k )Haar = k ! , (44)for d ≥ k . In this work, we restrict ourselves to studying moments of random circuits k < d = q n ,but note that the Haar value is modified once we probe moments greater than the dimension. TheHaar value of the frame potential arises from an old result on the moments of traces of randomunitaries [45]. When k > d , then | Tr( U ) | k is equal to the number of permutations in S k with thelongest increasing subsequence of length ≤ d [46].The difference in frame potentials is equal to the 2-norm distance between the moment operatorsas follows. Consider the operator∆ ≡ (cid:98) Φ ( k ) E − (cid:98) Φ ( k )Haar = (cid:90) E dU U ⊗ k ⊗ U †⊗ k − (cid:90) Haar dU U ⊗ k ⊗ U †⊗ k , (45)and note that Tr(∆ † ∆) = F ( k ) E − F ( k )Haar + F ( k )Haar = F ( k ) E − F ( k )Haar , (46)17here in the middle term we used the left/right invariance of the Haar measure to absorb the termsfrom the E average, giving the frame potential for the Haar ensemble. Moreover, as Tr(∆ † ∆) ≥ E , is lower bounded by the Haarvalue F ( k ) E ≥ F ( k )Haar as we claimed above. We now have that (cid:13)(cid:13)(cid:98) Φ ( k ) E − (cid:98) Φ ( k )Haar (cid:13)(cid:13) = F ( k ) E − F ( k )Haar . (47)As such, we can bound the diamond norm of the difference in k -fold channels in terms of theframe potentials of the two ensembles noting that (cid:13)(cid:13) Φ ( k ) E − Φ ( k )Haar (cid:13)(cid:13) (cid:5) ≤ d k (cid:13)(cid:13) Φ ( k ) E − Φ ( k )Haar (cid:13)(cid:13) → = d k (cid:13)(cid:13)(cid:98) Φ ( k ) E − (cid:98) Φ ( k )Haar (cid:13)(cid:13) ∞ ≤ d k (cid:13)(cid:13)(cid:98) Φ ( k ) E − (cid:98) Φ ( k )Haar (cid:13)(cid:13) , (48)and therefore (cid:13)(cid:13) Φ ( k ) E − Φ ( k )Haar (cid:13)(cid:13) (cid:5) ≤ d k (cid:0) F ( k ) E − F ( k )Haar (cid:1) . (49)Note that had we defined approximate designs in terms of the trace norm of the moment operators,the factors of d to bound in terms of the frame potential would be the same. Haar integration and Weingarten calculus
The general expression for integrating the k -th moment of U ( d ) is [47, 48] (cid:90) dU U i j . . . U i k j k U † (cid:96) m . . . U † (cid:96) k m k = (cid:88) σ,τ ∈ S k δ σ ( (cid:126)ı | (cid:126)m ) δ τ ( (cid:126) | (cid:126)(cid:96) ) W g ( σ − τ, d ) , (50)where we sum over elements of the permutation group S k and define a δ -function contractionindexed by a permutation σ as δ σ ( (cid:126)ı | (cid:126) ) = δ i ,j σ (1) . . . δ i k ,j σ ( k ) . The Weingarten function is a functionof a permutation σ ∈ S k and admits an expansion in terms of characters of the symmetric group as W g ( σ, d ) = 1 k ! (cid:88) λ (cid:96) k χ λ ( σ ) f λ c λ ( d ) , where c λ ( d ) = (cid:89) ( i,j ) ∈ λ ( d + j −
1) (51)and we sum over integer partitions of k , labelling the irreducible representations of S k , χ λ ( σ ) isan irreducible character of λ , and f λ is the dimension of the λ irrep. The polynomial c λ ( d ) is aproduct over the coordinates ( i, j ) of the Young diagram of the irrep λ . The Weingarten functionsare equivalently computed as the matrix inverse of the Gram matrix of permutation operators onthe k -fold space.Strictly speaking, the expression for the Weingarten function in Eq. 51 is valid for any d ≥ k ,such that the denominator is free of zeroes. For k > d , the expression is modified by taking thesum instead over integer partitions of length (cid:96) ( λ ) ≤ d . But we may drop this restriction and useEq. 51 for general moments (Prop 2.5 in [48]). W g ( σ, d ) is a rational function of d with finitelymany poles, but the expression for Haar integration in Eq. (50) will remain true for any d ∈ N using the full expression for the Weingarten function after some cancellation of poles. B Plaquette terms for higher k In Sec. 3, we explicitly derived the plaquette terms for k = 2, which we could interpret as simple rulesregarding the domain wall configurations separating regions of I ’s and S ’s in the lattice model. Butin order to discuss the nontrivial spin configurations contributing to the F ( k )RQC partition function,we need to understand the properties of plaquette terms for general k .18n this appendix, we will denote the plaquette terms with time running up, such that thetriangles are downward pointing as J σ σ σ = σ σ σ = (cid:88) τ ∈ S k W g ( σ − τ, q ) q (cid:96) ( τ − σ ) q (cid:96) ( τ − σ ) , (52)where again (cid:96) ( τ − σ ) denotes the length of the cycle-type of the permutation product; equivalent tothe number of closed loops in the product. For instance, if τ = σ − , then (cid:96) ( I ) = k as the cycle-typeof the identity is the partition into all ones.From now on in this appendix we will denote the plaquettes only in terms of the domainwalls representing transpositions between the permutations and drop the explicit dependence onpermutations. For instance, the empty plaquette corresponds to J σσσ , i.e. the permutationsdiffer by no transpositions. The plaquette corresponds to J σ σ σ , with σ differing from σ and σ by a single transposition, and σ and σ differing by two transpositions.Now we will prove some general properties of the plaquette terms for any k , which were pointedout in [16]. First, we want to show that plaquette terms evaluated on the same permutation J σσσ = 1for σ ∈ S k and J σσ (cid:48) σ (cid:48) = 0 when σ (cid:54) = σ (cid:48) . Recalling how the plaquette terms arise from the momentsof a single 2-site unitary U ⊗ k ⊗ U †⊗ k , the permutations σ and σ act on the outgoing indices of U and ingoing indices of U † as U ⊗ k σ σ U †⊗ k (53)Taking σ = σ = σ , we are simply contracting U ’s with U † ’s, which cancel and just give thepermutation operator on the k -fold space P σ . Therefore, plaquette terms with all the same per-mutation σ ∈ S k have weight one, and plaquettes with any number of ingoing domain walls and nooutgoing domain walls have weight zero = 1 and = 0 , (54)meaning that domain walls cannot simply annihilate. The same statement holds true for anynumber of incoming domain walls and no outgoing domain walls.Next, consider Eq. (53) with σ and σ differing by a single transposition, corresponding to asingle outgoing domain wall. This means that the k -fold unitaries U ⊗ k are contracted with theirdaggered counterparts and cancel on all but two of the k -fold unitaries, and we find UU U † U † I S = qq + 1 P I + qq + 1 P S , (55)on the two factors of the k -fold space H ⊗ k for which σ and σ differ by a transposition. Computingthe second moment above, we find that result is an identity or swap on the two factors of H ⊗ k forwhich σ and σ differ, i.e. σ can only differ from either σ or σ by a transposition. Thus wefind that plaquettes with only one outgoing domain wall must have only one ingoing domain wall,expressed diagrammatically as= = q q + 1 and = = 0 . (56) A permutation operator P σ permutes the k copies of H ⊗ k and acts on computational basis elements as P σ | i . . . i k (cid:105) = | i σ − (1) . . . i σ − ( k ) (cid:105) .
19e can also arrive at the same statements about the plaquette terms for general k from prop-erties of the Weingarten function [16], where W g ’s are the elements of the inverse of the matrix ofinner products of permutation operators. But this approach become subtle for k > d , when thematrix inverse is not strictly well-defined.Lastly, we note that the above discussion makes it clear that the plaquette rules for do notactually depend on k , but only on the factors for which σ and σ in Eq. (53) differ, which reducesto a moment calculation on those factors. For instance, if σ and σ differ by l transpositions,then the calculation of the J σ σ σ ’s will only involve computations of at most the 2 l -th moment, andpossibly less if the transpositions overlap. Plaquettes at large q Now we discuss the weights in the limit of large local dimension and show that an interaction termwith l ingoing domain walls contributes at most at order ∼ /q l . For the k -th moment, Weingartenfunctions have following asymptotic expansion in the dimension q [49] W g ( σ, q ) ∼ q k − (cid:96) ( σ )) , (57)with the sign determined by sgn( σ ). Precise expressions and bounds for the asymptotic form of W g were recently given in [50]. As the length of the cycle type is at most k , the W g ’s for σ ∈ S k contribute at order 1 /q k to 1 /q k . The longer the length of the cycle-type of σ , the higher orderthe contribution. At leading order in 1 /q , the interaction term J can be written J σ σ σ ∼ (cid:88) τ ∈ S k q (cid:96) ( τ − σ ) q (cid:96) ( τ − σ ) q k − (cid:96) ( σ − τ )) . (58)The length of the cycle type of σ is related to the number of transpositions as (cid:96) ( σ ) = k − | σ | , where | σ | is the minimal number of transpositions needed to write σ . The expression in the sum simplybecomes ( q | σ − τ | q | σ − τ | q | σ − τ | ) − .Given an interaction term with l ingoing domain walls, the term contributing at leading orderin the sum is τ = σ , corresponding to the leading Weingarten function. As σ differs from σ and σ by a total of l transpositions, the interaction contributes at order ∼ /q l . For an interactionterm with l ingoing and l outgoing domain walls, and thus no annihilating domain walls, this termis the only term contributing at leading order in 1 /q . But if we have domain wall annihilation inan interaction, other terms in the sum contribute at the same leading order. Take the simplestinteraction plaquette of this sort, with 4 ingoing and 2 outgoing domain walls. The τ = σ term contributes ∼ /q , but there is also a τ which differs from each permutation by a singletransposition, thus also contributing at ∼ /q but involving the next-to-leading order Weingartenfunction. As the sign of W g ( σ, d ) depends on the the signature of the permutation, plaquettes withannhilating domain walls can have negative weight. Explicit plaquette weights
Below we list some of the nontrivial plaquette terms for higher k , but we have computed all termsup to k = 6. Note that as the plaquette terms are symmetric in the outgoing permutations, all thetriangles weights below are reflection symmetric about the vertical axis.For k = 2, the only we only have a single domain wall corresponding to the swap transposition= qq + 1 . (59)20or k = 3, we have three transpositions and thus three types of domain walls, each colored differ-ently. The nonzero plaquette terms, also given in [16], are= qq + 1 , = q ( q − q + 2)( q + 1)( q − , = q ( q − − q + 2)( q + 1)( q − , = − q − q + 2)( q + 1)( q − , (60)as well as reflections and all possible colorings, e.g. there are three single domain wall terms foreach of the three transpositionsFor k = 4, the generating set for S consists of 6 transpositions. In addition to the sameplaquette terms for k = 3, there are 11 additional plaquette terms for k = 4 corresponding todifferent domain wall interactions. For instance, 4 of these terms can be represented as= ( q − q + 2 q + 2)( q + 3)( q + 2)( q + 1)( q − q , = q − q + 3)( q + 1)( q − , (61)= q − q − q + 3)( q + 1)( q − q , = − (2 q + 1)( q − q + 3)( q + 2)( q + 1)( q − q . Note that the terms with l ingoing domain walls contribute at order 1 /q l . We also note that noneof the plaquette terms have poles at physical values of the local dimension q ∈ N . This follows fromthe discussion in App. A, that the poles in the Weingarten function will cancel for physical valuesof the dimension [48]. We have computed all plaquette terms up to k = 6 and some additionalplaquette terms up to k = 8, and have verified both the expected asymptotic behavior in 1 /q andcancellation of poles at q ∈ N . References [1] A. W. Harrow and R. A. Low, “Random Quantum Circuits are Approximate 2-designs,”
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