Computational supplement

def pilot(K): h = math.ceil(math.log2(K+1)) cases = [("identity",0,0,0),("swap",0,0,K-1)] cases += [("entangling",L,seed,0) for L in (1,2) for seed in (0,1)] for family,L,seed,ell in cases: base = "K U = schedule(K,K,L,seed,base) for r in (F(1),F(1,2)): law = {0:[F(1)],**n:[1-r,r] for n in range(1,K+1)} entries = initial(localized(K,ell),K) pi = [F(0)]*K+[F(1)] for t in range(1,K+1): prev = pi entries = advance(entries,t,law,U,lambda j,r:1.) pi = popstep(pi,law) name = base+"_r A,dk,N,record = dashboard(entries,pi,K,U,t,name) record.update(case=name,K=K,L=L,seed=seed,t=t,r=str(r), family=family,internal_layers=t*L, spreading=sum(float(prev[n])*spreading( U.get((t,n),np.eye(2**n)),n) for n in range(K+1))) record["batch"] = recovery(N,name) record["streaming"] = {} for m in (0,1,2): out = (i,j):stream_decode(B,dk,m) for (i,j),B in N.items() record["streaming"][str(m)] = distance(out,name+"_m"+str(m)) for lab in sorted(labels(d) for d in dk): e = sum(map(len,lab)) ops = stream_ops(e,m) check(name+"_Kraus_ sum((Kx.conj().T@Kx for Kx,rec in ops), np.zeros((2**e,2**e),complex)), np.eye(2**e)) for Kx,rec in ops: DATA["occupancy"].append(stream_ledger(lab,m,h,rec)) outD = (i,j):collect_decode(B,dk,K,U) for (i,j),B in N.items() p = float(pi[0]) for i,j,E in units(): check(name+"_collect_ p*E+(1-p)*dep(E)) record["collect_error"] = .75*(1-p) for lab in sorted(labels(d) for d in dk): DATA["occupancy"].append(collect_ledger(lab,K,h,1)) DATA["records"].append(record) dump()

def absorption_audit(): K = 5 law = {0:[F(1)],**n:[1-F(1,2**n+1),F(1,2**n+1)] for n in range(1,K+1)} rs = F(1,33) need(F(K)/rs==165 and sum(1/law[n][1] for n in range(1,K+1))==67, "absorption means") pi = [F(0)]*K+[F(1)] for t in range(20): bound = sum(F(math.comb(t,i))*rs**i*(1-rs)**(t-i) for i in range(min(K-1,t)+1)) need(1-pi[0]<=bound and sum(pi)==1,"absorption tail") pi = popstep(pi,law)