diff --git a/science/bandwidth.md b/science/bandwidth.md index 2c29ed1..5bc2970 100644 --- a/science/bandwidth.md +++ b/science/bandwidth.md @@ -2,16 +2,16 @@ ## 1. Result, model, and comparison -There are three internally controllable regions: two modules L and R and a single bus qubit C. Each module contains N=n+s qubits: n designated payload qubits and s helper qubits. Assume n>=1 and s>=0. There is no direct L-R interaction. A pair interaction between any module qubit j and C has operator norm at most J. All contacts may act simultaneously. Arbitrary unitaries confined to a module or to C take zero time. The contact Hamiltonian may have arbitrary time dependence with a well-defined propagator; no minimum switching interval is imposed. Set hbar=1. +There are three internally controllable regions: two modules L and R and a single bus qubit C. Each module contains $`N=n+s`$ qubits: n designated payload qubits and s helper qubits. Assume $`n\ge1`$ and $`s\ge0`$. There is no direct L-R interaction. A pair interaction between any module qubit j and C has operator norm at most J. All contacts may act simultaneously. Arbitrary unitaries confined to a module or to C take zero time. The contact Hamiltonian may have arbitrary time dependence with a well-defined propagator; no minimum switching interval is imposed. Set hbar=1. The task is to exchange the two n-qubit payloads, including their correlations with arbitrary references. The construction returns C in its original arbitrary unknown state. Compare two helper specifications on **the same physical hardware and the same payload**: -- Clean: the s helpers in each module initially occupy |0>^s and must return there. +- Clean: the s helpers in each module initially occupy $`|0\rangle^{\otimes s}`$ and must return there. - Unknown: the s helpers contain arbitrary unknown quantum information, possibly entangled with references, and must be restored. There are no additional initialized degrees of freedom, measurements, resets, feedback, or input-dependent control sequences. No internal control connects different regions. In particular, the bus is not included in either freely controlled module. -Let T_clean(n,s) and T_unknown(n,s) be the infimum interaction times for the respective exact tasks. The main laws are +Let $`T_{\mathrm{clean}}(n,s)`$ and $`T_{\mathrm{unknown}}(n,s)`$ be the infimum interaction times for the respective exact tasks. The main laws are ```math \boxed{T_{\mathrm{clean}}(n,s)=\Theta\!\left(\frac{n}{J\sqrt{(n+s)(s+1)}}\right),} @@ -23,24 +23,24 @@ Let T_clean(n,s) and T_unknown(n,s) be the infimum interaction times for the res \qquad\text{(2)} ``` -The constants in these comparisons do not depend on n or s. Thus the optimal times at fixed n and s differ by Theta(sqrt(s+1)). This is not a comparison that gives the clean case more physical contacts. +The constants in these comparisons do not depend on n or s. Thus the optimal times at fixed n and s differ by $`\Theta(\sqrt{s+1})`$. This is not a comparison that gives the clean case more physical contacts. The bounds concern interaction time with free internal operations, not runtime with bounded local-gate speed. Preparation of the initial clean helpers is an assumed resource, not a charged operation. The bound is per contact, not a fixed total interaction-norm or control-work budget. Exact control lists can be exponentially long. These qualifications are part of the theorem, not implementation details to be omitted from it. -For 0<=s<=n, (1) simplifies to Theta(sqrt(n/(s+1))/J). The full formula also covers s>n. +For $`0\le s\le n`$, (1) simplifies to $`\Theta(\sqrt{n/(s+1)}/J)`$. The full formula also covers $`s\gt n`$. ## 2. Entropy deficit controls contact second moments The converse controls concentration in the contact spectrum through a conserved entropy deficit. -For a physical density operator omega on a D-dimensional Hilbert space, write +For a physical density operator $`\omega`$ on a D-dimensional Hilbert space, write ```math \mathcal I(\omega)=\log_2D-S(\omega). \qquad\text{(3)} ``` -This is an entropy deficit in bits. It is invariant under any unitary on the physical system. It is not the quadratic purity Tr omega^2 and is not an energy, work, or temperature. +This is an entropy deficit in bits. It is invariant under any unitary on the physical system. It is not the quadratic purity $`\mathrm{Tr}\omega^2`$ and is not an energy, work, or temperature. ### 2.1 Remove only cut-local terms @@ -58,7 +58,7 @@ Terms acting on only one side do not contribute to the entropy derivative and ar v_\mu^2=\sum_{j,a}c_{ja\mu}^2,\qquad v^2=\sum_\mu v_\mu^2. ``` -Orthogonal projection onto the two-sided Pauli coefficients contracts the normalized Frobenius norm. If the original edge has norm at most J_j, its normalized Frobenius norm is at most J_j. Therefore +Orthogonal projection onto the two-sided Pauli coefficients contracts the normalized Frobenius norm. If the original edge has norm at most $`J_j`$, its normalized Frobenius norm is at most $`J_j`$. Therefore ```math v^2\le\sum_jJ_j^2. @@ -69,7 +69,7 @@ This does NOT assume that subtracting the cut-local terms contracts the operator ### 2.2 A Gaussian bound on each collective coefficient -For fixed mu, the summands on distinct sites commute. With normalized trace tau and v_mu>0, +For fixed mu, the summands on distinct sites commute. With normalized trace tau and $`v_\mu\gt0`$, ```math \tau(e^{uA_\mu}) @@ -78,7 +78,7 @@ For fixed mu, the summands on distinct sites commute. With normalized trace tau \qquad\text{(6)} ``` -The scalar Gaussian integral identity e^{tx^2}=E_G exp(sqrt(2t)Gx), G standard normal, implies +The scalar Gaussian integral identity $`e^{tx^2}=\mathbb E_G e^{\sqrt{2t}Gx}`$, G standard normal, implies ```math \tau(e^{tA_\mu^2})\le(1-2tv_\mu^2)^{-1/2}, @@ -86,11 +86,11 @@ The scalar Gaussian integral identity e^{tx^2}=E_G exp(sqrt(2t)Gx), G standard n \qquad\text{(7)} ``` -These expressions are unchanged when A_mu is tensored with identities on the remaining physical qubits. +These expressions are unchanged when $`A_\mu`$ is tensored with identities on the remaining physical qubits. ### 2.3 Gibbs variational estimate at a fixed entropy deficit -Apply nonnegativity of relative entropy to omega and the normalized state exp(t A_mu^2)/Tr exp(t A_mu^2). Expanding this inequality, with natural logarithms in this calculation, gives +Apply nonnegativity of relative entropy to $`\omega`$ and the normalized state $`e^{tA_\mu^2}/\mathrm{Tr}(e^{tA_\mu^2})`$. Expanding this inequality, with natural logarithms in this calculation, gives ```math t\mathrm{Tr}(\omega A_\mu^2) @@ -98,7 +98,7 @@ t\mathrm{Tr}(\omega A_\mu^2) \qquad\text{(8)} ``` -No commutation between omega and A_mu is required. Choose t=1/(4v_mu^2). Equations (7)-(8) yield +No commutation between $`\omega`$ and $`A_\mu`$ is required. Choose $`t=1/(4v_\mu^2)`$. Equations (7)-(8) yield ```math \mathrm{Tr}(\omega A_\mu^2) @@ -106,15 +106,15 @@ No commutation between omega and A_mu is required. Choose t=1/(4v_mu^2). Equatio \qquad\text{(9)} ``` -The v_mu=0 case is trivial. +The $`v_\mu=0`$ case is trivial. -For Hermitian X_1,X_2,X_3, the operator inequality +For Hermitian $`X_1,X_2,X_3`$, the operator inequality ```math (X_1+X_2+X_3)^2\preceq3(X_1^2+X_2^2+X_3^2) ``` -follows because the difference is the sum over ib and f=0 otherwise; output leader q'=q xor f and output spectators r'=r xor q'. The collective magnetization of the output is (-1)^q |b-2w|. Thus a reversible permutation E obeys E^dagger(sum Z_j)E=Z_q D_b. Averaging all spectator XOR translations gives the scalar mean mu_b. +Each $`\epsilon_j`$ is an independent uniform sign. Here identity on spectators is a full operator identity, not an average channel on random spectator inputs. To recall the encoding explicitly, write a block basis label as (q,r), with spectator Hamming weight w. Set $`f=1`$ if $`2w\gt b`$ and $`f=0`$ otherwise; output leader $`q'=q\oplus f`$ and output spectators $`r'=r\oplus q'`$. The collective magnetization of the output is $`(-1)^q|b-2w|`$. Thus a reversible permutation E obeys $`E^\dagger(\sum_jZ_j)E=Z_qD_b`$. Averaging all spectator XOR translations gives the scalar mean $`\mu_b`$. All conjugated diagonal interactions commute. Equal-duration intervals therefore implement the averaged Hamiltonian **exactly**, with no vanishing-interval limit. Applying the block simulations simultaneously yields @@ -255,7 +255,7 @@ Before (19), perform the module-internal fanout F=\prod_{i=1}^{s}\mathrm{CNOT}_{q_0\to q_i}, ``` -and undo F afterward. On inputs with the helpers in |0>^s, +and undo F afterward. On inputs with the helpers in $`|0\rangle^{\otimes s}`$, ```math F^\dagger e^{-itH_{\mathrm{block}}}F @@ -277,13 +277,13 @@ The fanout creates and later removes correlations; it does not clone an unknown ### 5.3 Uniform gain estimate -For the sign sum S_b, E S_b^2=b and E S_b^4=3b^2-2b. Interpolation between its first, second, and fourth moments gives +For the sign sum $`S_b`$, $`\mathbb E S_b^2=b`$ and $`\mathbb E S_b^4=3b^2-2b`$. Interpolation between its first, second, and fourth moments gives ```math \sqrt{b/3}\le\mathbb E|S_b|=\mu_b\le\sqrt b. ``` -Balanced block sizes obey b_i>=N/(2m). Cauchy-Schwarz supplies the upper bound. Hence +Balanced block sizes obey $`b_i\ge N/(2m)`$. Cauchy-Schwarz supplies the upper bound. Hence ```math \boxed{\sqrt{(n+s)(s+1)/6}\le\gamma_{n,s}\le\sqrt{(n+s)(s+1)}.} @@ -294,9 +294,9 @@ No optimal prefactor is claimed. The explicit block grouping need not maximize t ### 5.4 Full payload exchange -Rotate the target and bus bases to realize the XX, YY, and ZZ versions of (20). The three angle-pi/4 operations commute on the target-bus subsystem and give SWAP up to a fixed global phase in time 3pi/(4J gamma_n,s). +Rotate the target and bus bases to realize the XX, YY, and ZZ versions of (20). The three $`\pi/4`$ operations commute on the target-bus subsystem and give SWAP up to a fixed global phase in time $`3\pi/(4J\gamma_{n,s})`$. -A buffered sequence uses 2n+1 target-bus SWAPs. For j=1,...,n, swap L_j with C, then R_j with C. Finally swap L_1 with C and rotate only the L payload labels left by one position. The output is precisely the payload exchange with the bus restored; all helpers were restored after each elementary gate. Therefore +A buffered sequence uses 2n+1 target-bus SWAPs. For j=1,...,n, swap $`L_j`$ with C, then $`R_j`$ with C. Finally swap $`L_1`$ with C and rotate only the L payload labels left by one position. The output is precisely the payload exchange with the bus restored; all helpers were restored after each elementary gate. Therefore ```math \boxed{T_{\mathrm{clean}}(n,s)\le\frac{3\pi(2n+1)}{4J\gamma_{n,s}}.} @@ -314,7 +314,7 @@ intervals per Ising gate. All conjugated contacts commute, so no short-time appr ## 6. Unknown helpers on identical hardware -For the unknown-helper task, put every physical module qubit, including the s helpers, in a Bell pair with its own inaccessible reference. The bus can be fixed pure for the lower bound. The physical density matrix always obeys omega<=2I/D. Orthogonality of the pair-contact Pauli coefficients gives +For the unknown-helper task, put every physical module qubit, including the s helpers, in a Bell pair with its own inaccessible reference. The bus can be fixed pure for the lower bound. The physical density matrix always obeys $`\omega\le2I/D`$. Orthogonality of the pair-contact Pauli coefficients gives ```math \mathrm{Tr}\omega H_{\mathrm{int}}^2\le2(n+s)J^2. @@ -327,7 +327,7 @@ T_{\mathrm{unknown}}(n,s)\ge\frac{\ln2}{2J}\frac{n}{\sqrt{n+s}}. \qquad\text{(24)} ``` -For the upper bound, use all N=n+s qubits in the occupied-data block amplifier. Its spectators, including the unknown helpers, are returned after each target-bus operation. Exchange only the n payload labels in the same buffered sequence: +For the upper bound, use all $`N=n+s`$ qubits in the occupied-data block amplifier. Its spectators, including the unknown helpers, are returned after each target-bus operation. Exchange only the n payload labels in the same buffered sequence: ```math T_{\mathrm{unknown}}(n,s) @@ -335,7 +335,7 @@ T_{\mathrm{unknown}}(n,s) \qquad\text{(25)} ``` -Since sqrt((n+s)/3)<=mu_(n+s)<=sqrt(n+s), (24)-(25) prove (2). Both cases have identical numbers and strengths of bus contacts, identical payloads, free internal controls, and helper return. The only altered promise is the initial helper state. +Since $`\sqrt{(n+s)/3}\le\mu_{n+s}\le\sqrt{n+s}`$, (24)-(25) prove (2). Both cases have identical numbers and strengths of bus contacts, identical payloads, free internal controls, and helper return. The only altered promise is the initial helper state. ## 7. Consequences and scope @@ -343,10 +343,10 @@ For s between zero and n, the exact optimal clean-helper interaction time has th | Clean helpers per module | Interaction time | |---|---| -| 0 or a fixed number | Theta(sqrt(n)/J) | -| Theta(log n) | Theta(sqrt(n/log n)/J) | -| Theta(sqrt n) | Theta(n^(1/4)/J) | -| Theta(n) | Theta(1/J) | +| 0 or a fixed number | $`\Theta(\sqrt n/J)`$ | +| $`\Theta(\log n)`$ | $`\Theta(\sqrt{n/\log n}/J)`$ | +| $`\Theta(\sqrt n)`$ | $`\Theta(n^{1/4}/J)`$ | +| $`\Theta(n)`$ | $`\Theta(1/J)`$ | The entries refer to ideal internal controls. In particular, the constant-time endpoint does not imply a constant-depth, bounded-strength implementation of the internal encodings or fanout. diff --git a/science/control-models.md b/science/control-models.md index 93d6ba0..83db4cd 100644 --- a/science/control-models.md +++ b/science/control-models.md @@ -20,29 +20,29 @@ The final permutation is unnecessary at $`n=1`$. Comparing only with an unamplif ### 2.1 Shorter three-data-qubit cycle -For q,a,b, the encoding E is Toffoli(a,b -> q), followed by CX(q,a) and CX(q,b). In the decoded frame, E^dagger (Z_q+Z_a+Z_b) E = Z_q diag(3,1,1,1). +For q,a,b, the encoding E is Toffoli(a,b -> q), followed by CX(q,a) and CX(q,b). In the decoded frame, $`E^\dagger(Z_q+Z_a+Z_b)E=Z_q\mathrm{diag}(3,1,1,1)`$. -Instead of repeating encode/evolve/decode four times, encode once, visit the spectator translations in Gray-code order 00,01,11,10, and decode once at the end. The physical transitions are E X_b E^dagger and E X_a E^dagger. They are Clifford affine permutations, although E itself is not Clifford. +Instead of repeating encode/evolve/decode four times, encode once, visit the spectator translations in Gray-code order 00,01,11,10, and decode once at the end. The physical transitions are $`EX_bE^\dagger`$ and $`EX_aE^\dagger`$. They are Clifford affine permutations, although E itself is not Clifford. For shift 01, a chronological circuit is -CX(q,a), CX(a,q), CX(a,b), CX(q,a) in parallel with X_b. +CX(q,a), CX(a,q), CX(a,b), CX(q,a) in parallel with $`X_b`$. For shift 10, it is -CX(q,b), CX(b,q), CX(b,a), CX(q,b) in parallel with X_a. +CX(q,b), CX(b,q), CX(b,a), CX(q,b) in parallel with $`X_a`$. -Each transition therefore takes 4delta. There are three transitions. After E^dagger at the end, an X_a restores the last Gray-code frame. +Each transition therefore takes $`4\delta`$. There are three transitions. After $`E^\dagger`$ at the end, an $`X_a`$ restores the last Gray-code frame. ### 2.2 No native Toffoli or initialized ancilla -Let V^2=X, with eigenvalues 1 and i. A standard exact five-two-qubit construction of Toffoli is +Let $`V^2=X`$, with eigenvalues 1 and i. A standard exact five-two-qubit construction of Toffoli is -CV(b,q), CX(a,b), CV^dagger(b,q), CX(a,b), CV(a,q). +CV(b,q), CX(a,b), $`\mathrm{CV}^\dagger(b,q)`$, CX(a,b), CV(a,q). -Its net exponent of V on q is b-(a xor b)+a=2ab. It is exact on arbitrary superpositions. +Its net exponent of V on q is $`b-(a\oplus b)+a=2ab`$. It is exact on arbitrary superpositions. -CX and X are Hermitian involutions and can be implemented, up to a common global phase, in delta=pi/(2G) with norm-G generators. Controlled V has spectral phases 0 and pi/2 and can be implemented in delta/2 with norm G after a scalar shift. Thus this Toffoli takes (7/2)delta and E takes (11/2)delta. The full cycle takes +CX and X are Hermitian involutions and can be implemented, up to a common global phase, in $`\delta=\pi/(2G)`$ with norm-G generators. Controlled V has spectral phases 0 and $`\pi/2`$ and can be implemented in $`\delta/2`$ with norm G after a scalar shift. Thus this Toffoli takes $`(7/2)\delta`$ and E takes $`(11/2)\delta`$. The full cycle takes ```math 2\cdot\frac{11}{2}\delta+3\cdot4\delta+\delta=24\delta @@ -52,7 +52,7 @@ of internal-control time, plus its charged contact time. This is an explicit fea ### 2.3 Recursive compilation and costs -For m=3^k data qubits assisting a designated target, the gain is a_k=(3/2)^k. Let D_k delta be the internal-control time for one amplified Ising operation. Disjoint triples are processed in parallel. Substitution of four lower-level simulations gives +For $`m=3^k`$ data qubits assisting a designated target, the gain is $`a_k=(3/2)^k`$. Let $`D_k\delta`$ be the internal-control time for one amplified Ising operation. Disjoint triples are processed in parallel. Substitution of four lower-level simulations gives ```math D_0=0,\qquad D_k=4D_{k-1}+24, @@ -60,7 +60,7 @@ D_0=0,\qquad D_k=4D_{k-1}+24, \qquad\text{(2)} ``` -There are 4^k raw contact intervals. The total number of elementary local gates (not depth) is +There are $`4^k`$ raw contact intervals. The total number of elementary local gates (not depth) is ```math N_k=30(12^k-1)/11. @@ -74,7 +74,7 @@ t_k=\frac{3\pi}{4J(3/2)^k}+ \qquad\text{(3)} ``` -With m<=n, a suitable set of m data qubits can assist each target in turn; they are restored after each completed operation. All within-register pair connectivity has been granted, so no extra movement to arrange triples is silently required. The full route takes +With $`m\le n`$, a suitable set of m data qubits can assist each target in turn; they are restored after each completed operation. All within-register pair connectivity has been granted, so no extra movement to arrange triples is silently required. The full route takes ```math \boxed{T_k=(2n+1)t_k+\pi/G\quad(n\gt 1).} @@ -85,16 +85,16 @@ The sequential option (1) is always included in optimization. The construction s ### 2.4 Finite-control consequence -Set R=G/J. Relative to the direct norm-J target-bus SWAP, Equation (3) becomes +Set $`R=G/J`$. Relative to the direct norm-J target-bus SWAP, Equation (3) becomes ```math \boxed{s_k(R)=(3/2)^{1-k}+\frac{24\,4^k-20}{R}.} \qquad\text{(5)} ``` -The k=1 primitive has s_1=1+76/R. It never beats the stronger direct-SWAP comparator under this compiler, even though its Ising coupling gain is 3/2. +The $`k=1`$ primitive has $`s_1=1+76/R`$. It never beats the stronger direct-SWAP comparator under this compiler, even though its Ising coupling gain is 3/2. -The k=2 block has s_2=2/3+364/R, which beats that comparator only for R>1092. This is a threshold for this feasible compiler, not a physical threshold or impossibility theorem for smaller R. +The $`k=2`$ block has $`s_2=2/3+364/R`$, which beats that comparator only for $`R\gt1092`$. This is a threshold for this feasible compiler, not a physical threshold or impossibility theorem for smaller R. Illustrative formula values, assuming the listed block fits inside n, are: @@ -108,7 +108,7 @@ Illustrative formula values, assuming the listed block fits inside n, are: These are ideal analytical timings, not experimental estimates or noisy fidelities. The required ratios are substantial, and this compiler is conservative. -For neighboring positive recursion levels, s_k=s_(k+1) at R=144*6^k. Thus once the preferred block fits, the optimal k depends on R, not on n. At fixed R, +For neighboring positive recursion levels, $`s_k=s_{k+1}`$ at $`R=144\cdot6^k`$. Thus once the preferred block fits, the optimal k depends on R, not on n. At fixed R, ```math \boxed{\min\{T_{\rm seq},T_1,T_2,\ldots\}=\Theta(n/J)} @@ -116,14 +116,14 @@ For neighboring positive recursion levels, s_k=s_(k+1) at R=144*6^k. Thus once t within this explicitly compiled family. -When R itself grows, balancing the two terms gives 6^k of order R and +When R itself grows, balancing the two terms gives $`6^k`$ of order R and ```math T_{\rm compiled}=O\left(\frac nJ R^{-\nu}\right), \qquad \nu=\frac{\ln(3/2)}{\ln6}=0.2262943855\ldots, ``` -in the regime where the optimal block is available. To use the whole m=n=3^k recursive register and retain the O(n^(log_3 2)/J) contact-time scaling, this compiler needs +in the regime where the optimal block is available. To use the whole $`m=n=3^k`$ recursive register and retain the $`O(n^{\log_3 2}/J)`$ contact-time scaling, this compiler needs ```math G/J\ \text{of order at least}\ n^{\log_3 6} @@ -136,20 +136,20 @@ up to constants. This is precisely the size-dependent control resource that the Finite duration does not erase every advantage. Combining the constructive timings with the [time-division converse](entropy-flow.md#5-no-helper-converse-and-sequential-comparator) gives a rigorous comparison against the whole time-division class, not merely one sequential schedule, when R is sufficiently large. -For example, at n=59049 and R=10^7, the best block in the compiled family is m=2187. Equations (3)-(4) give JT=23580.3126, whereas every time-division protocol satisfies JT>=40929.6479. No matrix of that size was simulated: these numbers are evaluations of the proved formulas. The huge control ratio, gate count, and ideal error-free assumptions make this a mathematical existence example, not a device proposal. It is a constant-factor comparison at fixed R, not asymptotic sublinear runtime. +For example, at $`n=59049`$ and $`R=10^7`$, the best block in the compiled family is $`m=2187`$. Equations (3)-(4) give $`JT=23580.3126`$, whereas every time-division protocol satisfies $`JT\ge40929.6479`$. No matrix of that size was simulated: these numbers are evaluations of the proved formulas. The huge control ratio, gate count, and ideal error-free assumptions make this a mathematical existence example, not a device proposal. It is a constant-factor comparison at fixed R, not asymptotic sublinear runtime. ## 3. Polynomial contact count at fixed uniform accuracy For the no-helper construction, the exponentially long exact list is not necessary for approximate routing in the same ideal internal-control model. This is a concentration argument about selecting a coherent control list, not a channel obtained by physically randomizing the input. -Select K independent, uniform spectator translations s_1,...,s_K OFFLINE, then fix that list for every run. The implemented diagonal coefficient is +Select K independent, uniform spectator translations $`s_1,\ldots,s_K`$ OFFLINE, then fix that list for every run. The implemented diagonal coefficient is ```math \bar D_K(r)=K^{-1}\sum_{\ell=1}^K D_n(r\oplus s_\ell). ``` -Every summand lies in [0,n] and has mean mu_n. Hoeffding's inequality followed by a union bound over the 2^(n-1) spectator labels gives +Every summand lies in [0,n] and has mean $`\mu_n`$. Hoeffding's inequality followed by a union bound over the $`2^{n-1}`$ spectator labels gives ```math \Pr\left(\|\bar D_K-\mu_nI\|\gt a\right) @@ -157,9 +157,9 @@ Every summand lies in [0,n] and has mean mu_n. Hoeffding's inequality followed b \qquad\text{(6)} ``` -All interval Hamiltonians commute exactly. An intended Ising angle theta=pi/4 therefore has full operator error at most theta a/mu_n. The entire route uses M=3(2n+1) such gates. Reusing a good list under target relabelings and basis conjugations, telescoping the product of unitaries bounds the final operator error by M theta a/mu_n, uniformly over all data and bus states. +All interval Hamiltonians commute exactly. An intended Ising angle $`\theta=\pi/4`$ therefore has full operator error at most $`\theta a/\mu_n`$. The entire route uses $`M=3(2n+1)`$ such gates. Reusing a good list under target relabelings and basis conjugations, telescoping the product of unitaries bounds the final operator error by $`M\theta a/\mu_n`$, uniformly over all data and bus states. -To obtain operator error at most eta with offline-list failure probability at most p, it suffices to choose +To obtain operator error at most $`\eta`$ with offline-list failure probability at most p, it suffices to choose ```math K\ge \frac{n^2M^2\theta^2}{2\eta^2\mu_n^2} @@ -167,7 +167,7 @@ K\ge \frac{n^2M^2\theta^2}{2\eta^2\mu_n^2} \qquad\text{(7)} ``` -For fixed p and eta, K=O(n^4/eta^2) per Ising gate and the total contact count is O(n^5/eta^2). The charged contact duration is unchanged. A full-unitary operator error eta implies channel diamond distance at most 2 eta. +For fixed p and $`\eta`$, $`K=O(n^4/\eta^2)`$ per Ising gate and the total contact count is $`O(n^5/\eta^2)`$. The charged contact duration is unchanged. A full-unitary operator error $`\eta`$ implies channel diamond distance at most 2 $`\eta`$. This gives both a probabilistic existence statement and a polynomial-time procedure for drawing a list with a rigorous high-probability uniform guarantee. It does not give an efficient verification of a large sampled list, a deterministic derandomization, an elementary-gate compilation of the global encoding E, or a practical sequence. The conservative constants are large; for small n the exact list is much shorter. A good list is fixed before the unknown input arrives and the physical execution is coherent: there is no measurement, postselection, or random mixing of quantum outputs. diff --git a/science/entropy-flow.md b/science/entropy-flow.md index d3a7432..d7c6f5a 100644 --- a/science/entropy-flow.md +++ b/science/entropy-flow.md @@ -2,7 +2,7 @@ ## 1. Published conditional-variance bound -Let rho_AB be an arbitrary finite-dimensional state. A may include arbitrarily many inaccessible reference systems. Let h act on AB, and let B have dimension d_B. Other Hamiltonian terms confined to A or to its complement do not contribute to dot S_A. Initially assume full rank; singular states are treated below. +Let $`\rho_{AB}`$ be an arbitrary finite-dimensional state. A may include arbitrarily many inaccessible reference systems. Let h act on AB, and let B have dimension $`d_B`$. Other Hamiltonian terms confined to A or to its complement do not contribute to $`\dot S_A`$. Initially assume full rank; singular states are treated below. The contact contribution can be written @@ -10,7 +10,7 @@ The contact contribution can be written \dot S_A=i\mathrm{Tr}\rho[h,\log_2\rho_A\otimes I_B], ``` -up to the irrelevant sign convention for the commutator. Since rho commutes with log rho, the same magnitude is obtained with the conditional modular difference K. Center both K and h, and apply Cauchy-Schwarz to the state-weighted inner product: +up to the irrelevant sign convention for the commutator. Since $`\rho`$ commutes with $`\log\rho`$, the same magnitude is obtained with the conditional modular difference K. Center both K and h, and apply Cauchy-Schwarz to the state-weighted inner product: ```math |\dot S_A| @@ -26,18 +26,18 @@ Using [DF19](../references/RELATED_WORK.md#df19), Corollary III.5, yields \qquad\text{(1)} ``` -For a bus qubit, the coefficient is c_DF=2 log_2 9. This combines the cited conditional-variance inequality with a state-weighted commutator estimate. +For a bus qubit, the coefficient is $`c_{\mathrm{DF}}=2\log_2 9`$. This combines the cited conditional-variance inequality with a state-weighted commutator estimate. ## 2. Elementary qubit constant -Use natural logarithms in this paragraph. Set sigma=rho_A tensor I_B and K=ln rho-ln sigma. Let r_i and s_j denote eigenvalues of rho and sigma, and use the spectral-overlap distribution +Use natural logarithms in this paragraph. Set $`\sigma=\rho_A\otimes I_B`$ and $`K=\ln\rho-\ln\sigma`$. Let $`r_i`$ and $`s_j`$ denote eigenvalues of $`\rho`$ and $`\sigma`$, and use the spectral-overlap distribution ```math p_{ij}=r_i|\langle i|j\rangle|^2, \qquad X_{ij}=\ln r_i-\ln s_j, ``` -including only r_i>0 and s_j>0. Then E X^2=Tr rho K^2; this identity is valid even when the logarithms do not commute. The operator inequality rho<=d_B sigma follows, for example, by pinching in a basis of B and bounding each positive diagonal block by rho_A. Let B_0=sigma^{-1/2}rho sigma^{-1/2} on the support of sigma. Then +including only $`r_i\gt0`$ and $`s_j\gt0`$. Then $`\mathbb E X^2=\mathrm{Tr}\rho K^2`$; this identity is valid even when the logarithms do not commute. The operator inequality $`\rho\le d_B\sigma`$ follows, for example, by pinching in a basis of B and bounding each positive diagonal block by $`\rho_A`$. Let $`B_0=\sigma^{-1/2}\rho\sigma^{-1/2}`$ on the support of $`\sigma`$. Then ```math \mathbb E e^X=\mathrm{Tr}\rho^2\sigma^{-1} @@ -50,7 +50,7 @@ and \mathbb E e^{-X}=\mathrm{Tr}\Pi_\rho\sigma\le d_B. ``` -The scalar inequality x^2<=e^x+e^{-x}-2 gives +The scalar inequality $`x^2\le e^x+e^{-x}-2`$ gives ```math \mathrm{Tr}\rho K^2\le2(d_B-1). @@ -64,19 +64,19 @@ Centering can only decrease this second moment. Therefore for a bus qubit, \qquad\text{(2)} ``` -This is a sharper qubit coefficient than (1), but does not change the exponent. The d_B dependence of this elementary bound is worse for large d_B than the published logarithmic estimate, so it is not an improved general-dimensional SIE theorem. +This is a sharper qubit coefficient than (1), but does not change the exponent. The $`d_B`$ dependence of this elementary bound is worse for large $`d_B`$ than the published logarithmic estimate, so it is not an improved general-dimensional SIE theorem. ## 3. Supports and admissible controls -When rho is singular, one must not project log sigma onto supp rho before squaring. Off-support components of log sigma acting on supp rho contribute to Tr rho K^2. One may assign finite values to log rho on its kernel, compute the rho-weighted moments on the support of sigma, and take a regularized limit. The rank-deficient verification cases use this convention. +When $`\rho`$ is singular, one must not project $`\log\sigma`$ onto $`\mathrm{supp}\rho`$ before squaring. Off-support components of $`\log\sigma`$ acting on $`\mathrm{supp}\rho`$ contribute to $`\mathrm{Tr}\rho K^2`$. One may assign finite values to $`\log\rho`$ on its kernel, compute the $`\rho`$-weighted moments on the support of $`\sigma`$, and take a regularized limit. The rank-deficient verification cases use this convention. For an integrated entropy inequality, regularize the entire finite physical/reference state by a small maximally mixed component, evolve it with the same unitary, prove the differential statement, integrate, and let the regularization vanish by entropy continuity. This avoids assuming differentiability at every rank-changing endpoint. -Internal pulses do not change S_A. General bounded-contact controls follow by their well-defined propagators and continuity; no Trotter step count enters the rate proof. Unbounded internal Hamiltonians are not included in the contact second moment because their entropy contribution across the chosen cut is identically zero. +Internal pulses do not change $`S_A`$. General bounded-contact controls follow by their well-defined propagators and continuity; no Trotter step count enters the rate proof. Unbounded internal Hamiltonians are not included in the contact second moment because their entropy contribution across the chosen cut is identically zero. ## 4. Weighted average-input entropy-flow theorem -Let L be any set of physical qubits, R the other data region, and C a set of bus qubits. There is no direct L-R coupling. For each c in C, write the genuinely interacting part of all L-c edges as +Let L be any set of physical qubits, R the other data region, and C a set of bus qubits. There is no direct L-R coupling. For each c in C, write the genuinely interacting part of all $`L\text{--}c`$ edges as ```math H_{Lc}^{\rm int}(t) @@ -84,7 +84,7 @@ H_{Lc}^{\rm int}(t) \gamma_{jcab}(t)\sigma_j^a\sigma_c^b. ``` -Terms acting on only one side are absorbed into internal controls. If the original edge h_jc has operator norm at most J_jc(t), orthogonal projection onto its two-sided Pauli terms contracts its normalized Frobenius norm. Distinct j,c,a,b strings are orthogonal, so +Terms acting on only one side are absorbed into internal controls. If the original edge $`h_{jc}`$ has operator norm at most $`J_{jc}(t)`$, orthogonal projection onto its two-sided Pauli terms contracts its normalized Frobenius norm. Distinct j,c,a,b strings are orthogonal, so ```math \|H_{Lc}^{\rm int}(t)\|_F^2 @@ -100,7 +100,7 @@ Consider a pure-state ensemble with \mathbb E_\psi |\psi\rangle\langle\psi|=I/D. ``` -Every input is acted on by the SAME input-independent controlled unitary. Its ensemble average remains I/D at all times, despite any correlations developed in individual members. Applying (2) separately to each c, averaging absolute values, and using Jensen gives +Every input is acted on by the SAME input-independent controlled unitary. Its ensemble average remains $`I/D`$ at all times, despite any correlations developed in individual members. Applying (2) separately to each c, averaging absolute values, and using Jensen gives ```math \begin{aligned} @@ -113,7 +113,7 @@ Every input is acted on by the SAME input-independent controlled unitary. Its en \qquad\text{(4)} ``` -The same theorem follows using published bound (1) with c_0 replaced by c_DF. Integrating proves +The same theorem follows using published bound (1) with $`c_0`$ replaced by $`c_{\mathrm{DF}}`$. Integrating proves ```math \boxed{\mathbb E_\psi|S_L(\psi_T)-S_L(\psi_0)| @@ -122,11 +122,11 @@ The same theorem follows using published bound (1) with c_0 replaced by c_DF. In \qquad\text{(5)} ``` -For uniform contacts, this is at most c_0 J |C| sqrt(|L|) T. There is no additive constant, factor n^delta, minimum segment length, or derivative condition. Compare the restricted-time-dependence average-input bound of [D26](../references/RELATED_WORK.md#d26). +For uniform contacts, this is at most $`c_0J|C|\sqrt{|L|}\,T`$. There is no additive constant, factor $`n^\delta`$, minimum segment length, or derivative condition. Compare the restricted-time-dependence average-input bound of [D26](../references/RELATED_WORK.md#d26). -This is not an input-uniform O(sqrt n) rate bound. For example, a GHZ logical qubit in the leaves under Z_C sum_j Z_j can have instantaneous entropy rate proportional to nJ. The proof neither assumes nor implies S(average rho)=average S(rho). It uses the linearity of the second moment and concavity of the square root. It also fails if the control schedule is selected using the unknown input's classical label: then the average evolved state need not stay maximally mixed. +This is not an input-uniform $`O(\sqrt n)`$ rate bound. For example, a GHZ logical qubit in the leaves under $`Z_C\sum_j Z_j`$ can have instantaneous entropy rate proportional to $`nJ`$. The proof neither assumes nor implies $`S(\mathbb E\rho)=\mathbb E S(\rho)`$. It uses the linearity of the second moment and concavity of the square root. It also fails if the control schedule is selected using the unknown input's classical label: then the average evolved state need not stay maximally mixed. -Additional known-initialized workspace is not free under this theorem. More generally, if an average physical state is bounded by kappa I/D, the same proof acquires sqrt(kappa). Unitary dynamics preserve that spectral domination. The entropy-deficit refinement and its matching workspace law are proved in [Initialization and routing bandwidth](bandwidth.md). +Additional known-initialized workspace is not free under this theorem. More generally, if an average physical state is bounded by $`\kappa I/D`$, the same proof acquires $`\sqrt\kappa`$. Unitary dynamics preserve that spectral domination. The entropy-deficit refinement and its matching workspace law are proved in [Initialization and routing bandwidth](bandwidth.md). ## 5. No-helper converse and sequential comparator