Datasets:

PureOne
/

NIRAEL-Quantum-Instant-Language

"""NIRAEL quantum-language checks and manuscript figures.
All data are finite synthetic calculations. The analytic proofs are in the PDF.
Requires Python 3.10+, NumPy, Matplotlib. No network access is used.
"""
from pathlib import Path
from itertools import combinations
from math import comb
import json
import numpy as np

ROOT = Path(__file__).resolve().parent
OUT = ROOT.parent / 'output' / 'pdf' / 'figures'
OUT.mkdir(parents=True, exist_ok=True)


def zeta_upper(a, n):
    a = np.array(a, float, copy=True)
    for i in range(n):
        bit = 1 << i
        for mask in range(1 << n):
            if not (mask & bit):
                a[mask] += a[mask | bit]
    return a


def walsh(a):
    v = np.array(a, float, copy=True)
    h = 1
    while h < len(v):
        for i in range(0, len(v), 2*h):
            left, right = v[i:i+h].copy(), v[i+h:i+2*h].copy()
            v[i:i+h], v[i+h:i+2*h] = left + right, left - right
        h *= 2
    return v


def periods(f):
    N = len(f)
    return [h for h in range(N) if all(f[x] == f[x ^ h] for x in range(N))]


def check_periods():
    n = 3; N = 1 << n
    checked = 0
    rng = np.random.default_rng(260624)
    for fm in range(1 << N):
        f = np.array([(fm >> x) & 1 for x in range(N)])
        H = periods(f)
        annihilator = [a for a in range(N) if all((a & h).bit_count() % 2 == 0 for h in H)]
        p = np.zeros(N)
        p[annihilator] = 1/len(annihilator)
        w = walsh(p)
        assert max((abs(w[x ^ y]) for x in range(N) for y in range(N) if f[x] != f[y]), default=0) < 1e-12
        # A translation-invariant randomized mixture of annihilator cosets.
        base = rng.random(N); base /= base.sum()
        q = sum(base[np.arange(N) ^ a] for a in annihilator)/len(annihilator)
        wq = walsh(q)
        assert all(abs(wq[h]) < 1e-12 for h in range(N) if h not in H)
        checked += 1
    hamming = []
    for n in range(1, 9):
        for k in range(n):
            f = np.array([int(x.bit_count() <= k) for x in range(1 << n)])
            assert periods(f) == [0]
            hamming.append((n,k))
    return {'boolean_functions_checked': checked, 'hamming_ball_predicates_checked':len(hamming)}


def rank_f2(a):
    a=np.array(a,dtype=np.uint8,copy=True)
    if not a.size: return 0
    r=0
    for c in range(a.shape[1]):
        piv=np.flatnonzero(a[r:,c])
        if not len(piv): continue
        j=r+int(piv[0]); a[[r,j]]=a[[j,r]]
        for i in range(a.shape[0]):
            if i!=r and a[i,c]: a[i]^=a[r]
        r+=1
        if r==a.shape[0]: break
    return r


def reduced_betti(faces):
    # Augmented simplicial complex, over F_2. Empty face included.
    levels={d:[f for f in faces if f.bit_count()==d+1] for d in range(-1,6)}
    ranks={}
    for d in range(0,6):
        rows=levels[d-1]; cols=levels[d]
        mat=np.zeros((len(rows),len(cols)),dtype=np.uint8)
        lookup={r:i for i,r in enumerate(rows)}
        for j,f in enumerate(cols):
            for i in range(5):
                if f & (1<<i): mat[lookup[f^(1<<i)],j]=1
        ranks[d]=rank_f2(mat)
    return {d:len(levels[d])-ranks.get(d,0)-ranks.get(d+1,0) for d in range(-1,5)}


def check_homology():
    n=5; forbidden=[0b00111,0b11100,0b01001]
    faces=[f for f in range(1<<n) if not any(f&c==c for c in forbidden)]
    coeff=np.zeros(1<<n,dtype=int)
    betti=[]
    for S in range(1<<n):
        induced=[f for f in faces if f&S==f]
        bs=reduced_betti(induced)
        for i in range(n+1):
            val=bs.get(S.bit_count()-i-1,0)
            if val:
                betti.append({'homological_degree':i,'support':S,'value':val})
                coeff[S]+=(-1)**i*val
    expected=np.zeros(1<<n,dtype=int)
    expected[0]=1
    for mask in forbidden: expected[mask]-=1
    expected[0b01111]+=1
    expected[0b11101]+=1
    assert np.array_equal(coeff,expected)
    rng=np.random.default_rng(413)
    error=0
    for _ in range(100):
        q=rng.random(n)
        accept=sum(np.prod([q[i] if f&(1<<i) else 1-q[i] for i in range(n)]) for f in faces)
        algebra=sum(coeff[S]*np.prod([q[i] for i in range(n) if S&(1<<i)]) for S in range(1<<n))
        error=max(error,abs(accept-algebra))
    assert error<1e-12
    return {'faces':len(faces),'minimal_nonfaces':forbidden,'nonzero_betti':betti,
            'hilbert_numerator_coefficients':{str(i):int(c) for i,c in enumerate(coeff) if c},
            'max_probability_error':float(error),'all_active_q_0_64_refusal':float(.64**2+2*.64**3-2*.64**4)}


def check_compiler():
    n=8; N=1<<n
    candidates=np.array([sum(1<<i for i in C) for C in combinations(range(n),3)])
    rng=np.random.default_rng(20261024)
    # A frozen nonuniform prior with two clusters of frequent deviations.
    x=np.arange(N)
    prior=rng.random(N)**2
    prior*=1+4*((x&7)==7)+3*((x&224)==224)
    prior/=prior.sum()
    local_q=np.array([.88,.82,.73,.92,.62,.79,.84,.68])
    gamma=np.array([np.prod([local_q[i] for i in range(n) if c&(1<<i)]) for c in candidates])
    r=np.array([g*((x&c)==c) for c,g in zip(candidates,gamma)])
    v=prior.copy(); chosen=[]; scores=[0.0]; maxerr=0
    for _ in range(4):
        z=zeta_upper(v,n)
        gain=gamma*z[candidates]
        direct=r@v
        maxerr=max(maxerr,float(np.max(abs(gain-direct))))
        gain[chosen]=-1
        j=int(np.argmax(gain)); chosen.append(j)
        v*=1-r[j]
        scores.append(float(1-v.sum()))
    exact_value=-1; exact=None
    # Three packets allow a meaningful complete finite optimizer (27,720 choices).
    for S in combinations(range(len(candidates)),3):
        val=float(prior@(1-np.prod(1-r[list(S)],axis=0)))
        if val>exact_value: exact_value=val; exact=S
    violations=0; min_slack=1
    for _ in range(1500):
        p=rng.permutation(len(candidates))
        A=p[:2]; B=p[:5]; j=p[6]
        ga=float(prior@(r[j]*np.prod(1-r[A],axis=0)))
        gb=float(prior@(r[j]*np.prod(1-r[B],axis=0)))
        min_slack=min(min_slack,ga-gb)
        violations+=int(ga+1e-12<gb)
    assert maxerr<1e-12 and violations==0
    return {'n':n,'candidates':len(candidates),'seed':20261024,
            'greedy_packets':[int(candidates[j]) for j in chosen], 'greedy_success':scores,
            'exact_three_packet_success':exact_value,
            'exact_three_packets':[int(candidates[j]) for j in exact],
            'three_packet_greedy_ratio':scores[3]/exact_value,
            'sets_exhaustively_checked':comb(len(candidates),3),
            'zeta_direct_max_error':maxerr,'submodularity_checks':1500,
            'submodularity_violations':violations,'minimum_diminishing_return_slack':min_slack}


def make_figures(report):
    import matplotlib
    matplotlib.use('Agg')
    import matplotlib.pyplot as plt
    from check_local_global import product_gram
    plt.rcParams.update({'font.family':'DejaVu Sans','font.size':10,'axes.spines.top':False,
                         'axes.spines.right':False,'axes.labelcolor':'#14222B','text.color':'#14222B',
                         'figure.facecolor':'white','axes.prop_cycle':plt.cycler(color=['#187F70','#AD5269','#6C7890'])})
    ss=np.linspace(.015,.985,110)
    fig,axs=plt.subplots(1,2,figsize=(9,3.1),layout='constrained')
    for ax,n,kind in [(axs[0],2,'AND'),(axs[1],3,'MAJORITY')]:
        f=np.array([int(x==2**n-1) if kind=='AND' else int(x.bit_count()>n//2) for x in range(2**n)])
        d=1-2*f; local=[]; glob=[]
        for s in ss:
            _,R=product_gram([s]*n)
            H=(R*d[None,:])@R/(2**n)
            glob.append((1+np.sum(abs(np.linalg.eigvalsh(H))))/2)
            local.append((1+np.trace(d[:,None]*H))/2)
        ax.plot(ss,glob,label='Optimal collective readout',lw=2)
        ax.plot(ss,local,label='Bitwise Helstrom, then f',lw=1.8)
        ax.fill_between(ss,local,glob,color='#73E2C0',alpha=.25)
        ax.set(title=f'{kind} on {n} bits',xlabel='Pure-state overlap s',ylabel='Success probability',ylim=(.48,1.01))
    axs[0].legend(fontsize=8)
    fig.savefig(OUT/'local_global_boundary.pdf'); plt.close(fig)
    fig,axs=plt.subplots(1,2,figsize=(9,3.1),layout='constrained')
    q=np.linspace(0,1,301)
    for w in [3,5,8]:
        vals=sum(comb(w,j)*q**j*(1-q)**(w-j) for j in range(3,w+1))
        axs[0].plot(q,vals,label=f'{w} active symbols',lw=1.8)
    axs[0].set(xlabel='Local conclusive activation q',ylabel='Optimal refusal probability',title='Refuse the at-most-two grammar')
    axs[0].legend(fontsize=8)
    axs[1].plot(q,q**2+2*q**3-2*q**4,lw=2,label='Three overlapping forbidden sets')
    axs[1].plot(q,q**3,lw=1.8,label='One three-symbol certificate')
    axs[1].set(xlabel='Local conclusive activation q',ylabel='Optimal refusal probability',title='Grammar changes the response law')
    axs[1].legend(fontsize=8)
    fig.savefig(OUT/'grammar_reliability.pdf'); plt.close(fig)
    c=report['compiler']; fig,ax=plt.subplots(figsize=(7,2.7),layout='constrained')
    ax.plot(range(5),c['greedy_success'],'o-',color='#187F70',lw=2,label='Greedy complete-certificate packets')
    ax.scatter([3],[c['exact_three_packet_success']],marker='D',color='#AD5269',s=55,label='Exhaustive optimum at three packets',zorder=3)
    ax.set(xlabel='Independent packet budget',ylabel='Expected sound refusal',xticks=range(5),ylim=(0,.55))
    ax.legend(fontsize=8)
    fig.savefig(OUT/'certificate_compiler.pdf'); plt.close(fig)


def main():
    report={'status':'finite synthetic verification, not experimental quantum hardware',
            'periods':check_periods(),'homology':check_homology(),'compiler':check_compiler()}
    (ROOT/'nirael_results.json').write_text(json.dumps(report,indent=2))
    make_figures(report)
    print(json.dumps(report,indent=2))


if __name__=='__main__': main()

	"""NIRAEL quantum-language checks and manuscript figures.

	All data are finite synthetic calculations. The analytic proofs are in the PDF.
	Requires Python 3.10+, NumPy, Matplotlib. No network access is used.
	"""
	from pathlib import Path
	from itertools import combinations
	from math import comb
	import json
	import numpy as np

	ROOT = Path(__file__).resolve().parent
	OUT = ROOT.parent / 'output' / 'pdf' / 'figures'
	OUT.mkdir(parents=True, exist_ok=True)


	def zeta_upper(a, n):
	a = np.array(a, float, copy=True)
	for i in range(n):
	bit = 1 << i
	for mask in range(1 << n):
	if not (mask & bit):
	a[mask] += a[mask \| bit]
	return a


	def walsh(a):
	v = np.array(a, float, copy=True)
	h = 1
	while h < len(v):
	for i in range(0, len(v), 2*h):
	left, right = v[i:i+h].copy(), v[i+h:i+2*h].copy()
	v[i:i+h], v[i+h:i+2*h] = left + right, left - right
	h *= 2
	return v


	def periods(f):
	N = len(f)
	return [h for h in range(N) if all(f[x] == f[x ^ h] for x in range(N))]


	def check_periods():
	n = 3; N = 1 << n
	checked = 0
	rng = np.random.default_rng(260624)
	for fm in range(1 << N):
	f = np.array([(fm >> x) & 1 for x in range(N)])
	H = periods(f)
	annihilator = [a for a in range(N) if all((a & h).bit_count() % 2 == 0 for h in H)]
	p = np.zeros(N)
	p[annihilator] = 1/len(annihilator)
	w = walsh(p)
	assert max((abs(w[x ^ y]) for x in range(N) for y in range(N) if f[x] != f[y]), default=0) < 1e-12
	# A translation-invariant randomized mixture of annihilator cosets.
	base = rng.random(N); base /= base.sum()
	q = sum(base[np.arange(N) ^ a] for a in annihilator)/len(annihilator)
	wq = walsh(q)
	assert all(abs(wq[h]) < 1e-12 for h in range(N) if h not in H)
	checked += 1
	hamming = []
	for n in range(1, 9):
	for k in range(n):
	f = np.array([int(x.bit_count() <= k) for x in range(1 << n)])
	assert periods(f) == [0]
	hamming.append((n,k))
	return {'boolean_functions_checked': checked, 'hamming_ball_predicates_checked':len(hamming)}


	def rank_f2(a):
	a=np.array(a,dtype=np.uint8,copy=True)
	if not a.size: return 0
	r=0
	for c in range(a.shape[1]):
	piv=np.flatnonzero(a[r:,c])
	if not len(piv): continue
	j=r+int(piv[0]); a[[r,j]]=a[[j,r]]
	for i in range(a.shape[0]):
	if i!=r and a[i,c]: a[i]^=a[r]
	r+=1
	if r==a.shape[0]: break
	return r


	def reduced_betti(faces):
	# Augmented simplicial complex, over F_2. Empty face included.
	levels={d:[f for f in faces if f.bit_count()==d+1] for d in range(-1,6)}
	ranks={}
	for d in range(0,6):
	rows=levels[d-1]; cols=levels[d]
	mat=np.zeros((len(rows),len(cols)),dtype=np.uint8)
	lookup={r:i for i,r in enumerate(rows)}
	for j,f in enumerate(cols):
	for i in range(5):
	if f & (1<<i): mat[lookup[f^(1<<i)],j]=1
	ranks[d]=rank_f2(mat)
	return {d:len(levels[d])-ranks.get(d,0)-ranks.get(d+1,0) for d in range(-1,5)}


	def check_homology():
	n=5; forbidden=[0b00111,0b11100,0b01001]
	faces=[f for f in range(1<<n) if not any(f&c==c for c in forbidden)]
	coeff=np.zeros(1<<n,dtype=int)
	betti=[]
	for S in range(1<<n):
	induced=[f for f in faces if f&S==f]
	bs=reduced_betti(induced)
	for i in range(n+1):
	val=bs.get(S.bit_count()-i-1,0)
	if val:
	betti.append({'homological_degree':i,'support':S,'value':val})
	coeff[S]+=(-1)*ival
	expected=np.zeros(1<<n,dtype=int)
	expected[0]=1
	for mask in forbidden: expected[mask]-=1
	expected[0b01111]+=1
	expected[0b11101]+=1
	assert np.array_equal(coeff,expected)
	rng=np.random.default_rng(413)
	error=0
	for _ in range(100):
	q=rng.random(n)
	accept=sum(np.prod([q[i] if f&(1<<i) else 1-q[i] for i in range(n)]) for f in faces)
	algebra=sum(coeff[S]*np.prod([q[i] for i in range(n) if S&(1<<i)]) for S in range(1<<n))
	error=max(error,abs(accept-algebra))
	assert error<1e-12
	return {'faces':len(faces),'minimal_nonfaces':forbidden,'nonzero_betti':betti,
	'hilbert_numerator_coefficients':{str(i):int(c) for i,c in enumerate(coeff) if c},
	'max_probability_error':float(error),'all_active_q_0_64_refusal':float(.64*2+2.64*3-2.64**4)}


	def check_compiler():
	n=8; N=1<<n
	candidates=np.array([sum(1<<i for i in C) for C in combinations(range(n),3)])
	rng=np.random.default_rng(20261024)
	# A frozen nonuniform prior with two clusters of frequent deviations.
	x=np.arange(N)
	prior=rng.random(N)**2
	prior=1+4((x&7)==7)+3*((x&224)==224)
	prior/=prior.sum()
	local_q=np.array([.88,.82,.73,.92,.62,.79,.84,.68])
	gamma=np.array([np.prod([local_q[i] for i in range(n) if c&(1<<i)]) for c in candidates])
	r=np.array([g*((x&c)==c) for c,g in zip(candidates,gamma)])
	v=prior.copy(); chosen=[]; scores=[0.0]; maxerr=0
	for _ in range(4):
	z=zeta_upper(v,n)
	gain=gamma*z[candidates]
	direct=r@v
	maxerr=max(maxerr,float(np.max(abs(gain-direct))))
	gain[chosen]=-1
	j=int(np.argmax(gain)); chosen.append(j)
	v*=1-r[j]
	scores.append(float(1-v.sum()))
	exact_value=-1; exact=None
	# Three packets allow a meaningful complete finite optimizer (27,720 choices).
	for S in combinations(range(len(candidates)),3):
	val=float(prior@(1-np.prod(1-r[list(S)],axis=0)))
	if val>exact_value: exact_value=val; exact=S
	violations=0; min_slack=1
	for _ in range(1500):
	p=rng.permutation(len(candidates))
	A=p[:2]; B=p[:5]; j=p[6]
	ga=float(prior@(r[j]*np.prod(1-r[A],axis=0)))
	gb=float(prior@(r[j]*np.prod(1-r[B],axis=0)))
	min_slack=min(min_slack,ga-gb)
	violations+=int(ga+1e-12<gb)
	assert maxerr<1e-12 and violations==0
	return {'n':n,'candidates':len(candidates),'seed':20261024,
	'greedy_packets':[int(candidates[j]) for j in chosen], 'greedy_success':scores,
	'exact_three_packet_success':exact_value,
	'exact_three_packets':[int(candidates[j]) for j in exact],
	'three_packet_greedy_ratio':scores[3]/exact_value,
	'sets_exhaustively_checked':comb(len(candidates),3),
	'zeta_direct_max_error':maxerr,'submodularity_checks':1500,
	'submodularity_violations':violations,'minimum_diminishing_return_slack':min_slack}


	def make_figures(report):
	import matplotlib
	matplotlib.use('Agg')
	import matplotlib.pyplot as plt
	from check_local_global import product_gram
	plt.rcParams.update({'font.family':'DejaVu Sans','font.size':10,'axes.spines.top':False,
	'axes.spines.right':False,'axes.labelcolor':'#14222B','text.color':'#14222B',
	'figure.facecolor':'white','axes.prop_cycle':plt.cycler(color=['#187F70','#AD5269','#6C7890'])})
	ss=np.linspace(.015,.985,110)
	fig,axs=plt.subplots(1,2,figsize=(9,3.1),layout='constrained')
	for ax,n,kind in [(axs[0],2,'AND'),(axs[1],3,'MAJORITY')]:
	f=np.array([int(x==2n-1) if kind=='AND' else int(x.bit_count()>n//2) for x in range(2n)])
	d=1-2*f; local=[]; glob=[]
	for s in ss:
	_,R=product_gram([s]*n)
	H=(Rd[None,:])@R/(2*n)
	glob.append((1+np.sum(abs(np.linalg.eigvalsh(H))))/2)
	local.append((1+np.trace(d[:,None]*H))/2)
	ax.plot(ss,glob,label='Optimal collective readout',lw=2)
	ax.plot(ss,local,label='Bitwise Helstrom, then f',lw=1.8)
	ax.fill_between(ss,local,glob,color='#73E2C0',alpha=.25)
	ax.set(title=f'{kind} on {n} bits',xlabel='Pure-state overlap s',ylabel='Success probability',ylim=(.48,1.01))
	axs[0].legend(fontsize=8)
	fig.savefig(OUT/'local_global_boundary.pdf'); plt.close(fig)
	fig,axs=plt.subplots(1,2,figsize=(9,3.1),layout='constrained')
	q=np.linspace(0,1,301)
	for w in [3,5,8]:
	vals=sum(comb(w,j)qj(1-q)**(w-j) for j in range(3,w+1))
	axs[0].plot(q,vals,label=f'{w} active symbols',lw=1.8)
	axs[0].set(xlabel='Local conclusive activation q',ylabel='Optimal refusal probability',title='Refuse the at-most-two grammar')
	axs[0].legend(fontsize=8)
	axs[1].plot(q,q*2+2q*3-2q**4,lw=2,label='Three overlapping forbidden sets')
	axs[1].plot(q,q**3,lw=1.8,label='One three-symbol certificate')
	axs[1].set(xlabel='Local conclusive activation q',ylabel='Optimal refusal probability',title='Grammar changes the response law')
	axs[1].legend(fontsize=8)
	fig.savefig(OUT/'grammar_reliability.pdf'); plt.close(fig)
	c=report['compiler']; fig,ax=plt.subplots(figsize=(7,2.7),layout='constrained')
	ax.plot(range(5),c['greedy_success'],'o-',color='#187F70',lw=2,label='Greedy complete-certificate packets')
	ax.scatter([3],[c['exact_three_packet_success']],marker='D',color='#AD5269',s=55,label='Exhaustive optimum at three packets',zorder=3)
	ax.set(xlabel='Independent packet budget',ylabel='Expected sound refusal',xticks=range(5),ylim=(0,.55))
	ax.legend(fontsize=8)
	fig.savefig(OUT/'certificate_compiler.pdf'); plt.close(fig)


	def main():
	report={'status':'finite synthetic verification, not experimental quantum hardware',
	'periods':check_periods(),'homology':check_homology(),'compiler':check_compiler()}
	(ROOT/'nirael_results.json').write_text(json.dumps(report,indent=2))
	make_figures(report)
	print(json.dumps(report,indent=2))


	if __name__=='__main__': main()