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Quantum Spectral Estimation and Matrix Functions — Examples

1 · Prerequisites

2 · Summary

The first example computes a two-eigenvalue state distribution and gives a finite QPE resolution and controlled-query count. The second constructs an exact one-ancilla dilation, separates cosine and negative sine into even and odd QSVT polynomials, and checks the normalized exponential error at both eigenvalues.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passOpen item page →

Spectral distribution and phase resolution for a two-level matrix

Example

Take $H = diag (- 1 / 2, 1 / 3)$ on $C^{2}$ and $ψ = \sqrt{3 / 5} ∣ 0 ⟩ + \sqrt{2 / 5} ∣ 1 ⟩$ . The state-dependent spectral distribution has masses $3 / 5$ at $- 1 / 2$ and $2 / 5$ at $1 / 3$ . With $U = e^{i H}$ and $Q = 128$ , the measured phase law is the mixture of the two exact QPE kernels with those weights, at the canonical phase representatives $1 - 1 / (4 π)$ and $1 / (6 π)$ . For additive eigenvalue accuracy $ϵ = 1 / 4$ , the total probability of an estimate within $1 / 4$ of its eigenvalue is at least $3 / 4$ ; the two success intervals are disjoint. With only controlled- $U$ access, the phase register uses $127$ controlled- $U$ calls per run.

Facts & Assumptions

Given: The diagonal matrix, normalized input, exact QPE access to $U = e^{i H}$ , and phase-register size $m = 7$ ( $Q = 2^{7} = 128$ ).

[F1]

State-dependent spectral distribution and QPE output law and conditional state for a spectral input: each eigenspace has weight $∥ P_{λ} ψ ∥^{2}$ and the sampler law is the corresponding mixture of phase kernels.

[F2]

Phase-estimation nearest-grid and tail bounds: if $K = ⌊ Q η ⌋ \geq 2$ , QPE has circular phase-error tail at most $1 / (2 (K - 1))$ .

[F3]

Pi is the first positive zero of sine, Sine is positive and cosine is strictly decreasing on (0,2), with cos 2 at most -1/3, and The exponential definitions of complex sine, cosine, hyperbolic sine, and hyperbolic cosine equal their entire power series: $\sin π = 0$ , sine is positive on $(0, π)$ , $π > 2$ because sine has no zero on $(0, 2]$ , and sine has its stated Taylor series.

[F4]

The alternating series test: if $(b_{k})$ is nonincreasing with $b_{k} \to 0$ then $\sum_{k} (- 1)^{k} b_{k}$ converges, the sum lies between any two consecutive partial sums, and the error after $n$ terms is at most $b_{n}$ : for an alternating series with nonincreasing nonnegative term magnitudes tending to zero, the series converges and the remainder after a partial sum is bounded by the first omitted term.

[F5]

Eigenvalue-estimation access and aliasing convention: with only controlled- $U$ access, implementing powers $U^{2^{j}}$ by repeated calls costs $\sum_{j = 0}^{m - 1} 2^{j} = 2^{m} - 1$ controlled- $U$ calls.

Verification

1.1F1givenalgebra

The two diagonal entries are distinct, so their spectral projections are $∣ 0 ⟩ ⟨ 0 ∣$ and $∣ 1 ⟩ ⟨ 1 ∣$ . Their action on $ψ$ gives squared norms $3 / 5$ and $2 / 5$ , respectively; the squares sum to one. The encoded phase classes have canonical representatives $1 - 1 / (4 π)$ for eigenvalue $- 1 / 2$ and $1 / (6 π)$ for eigenvalue $1 / 3$ . By [F1], the squared norms are exactly the masses and the measured phase law is $\frac{3}{5} P_{1 - 1 / (4 π)} + \frac{2}{5} P_{1 / (6 π)}$ .

1.2F3F4givenalgebra

We certify $π < 4$ . The sine Taylor partial sum through $- 4^{7} / 7!$ is $S_{3} = 4 - 4^{3} / 3! + 4^{5} / 5! - 4^{7} / 7! = - 6976 / 5040 = - 502272 / 362880$ . Starting with $4^{9} / 9!$ , the absolute Taylor terms decrease because the next-to-current ratio is $16 / ((n + 1) (n + 2)) < 1$ for odd $n \geq 7$ . These magnitudes tend to zero, since the sine series at $4$ converges by [F3]. Applying [F4] to this tail bounds the alternating remainder by $4^{9} / 9! = 262144 / 362880$ . Hence $\sin 4 \leq - 240128 / 362880 < 0$ . If $π > 4$ , [F3] would give $\sin 4 > 0$ ; if $π = 4$ , [F3] would give $\sin 4 = 0$ . Both contradict the negative bound, so $π < 4$ .

2.1F1F2F3step 1.2givenalgebra

Put $η = ϵ / (2 π) = 1 / (8 π)$ . For each eigenvalue $λ$ , let $θ_{0} = λ / (2 π)$ and let $\overset{ˉ}{θ} \in [0, 1)$ be its canonical encoded phase. Then $Q η = 16 / π > 4$ by $π < 4$ , so $K = ⌊ Q η ⌋ \geq 4$ ; [F2] bounds the circular tail at $\overset{ˉ}{θ}$ by at most $1 / 6 < 1 / 4$ . On this success event, the circular-distance definition gives an integer $k$ with $∣ y / Q + k - \overset{ˉ}{θ} ∣ < η$ . Writing $\overset{ˉ}{θ} = θ_{0} + n$ for an integer $n$ , the lift $y / Q + k - n = θ_{0} + e$ has $∣ e ∣ < η$ . Since $∣ θ_{0} ∣ \leq 1 / (2 π)$ , we have $∣ θ_{0} + e ∣ < 5 / (8 π) < 1 / 2$ by $π > 2$ , so it is the principal representative and the decoded eigenvalue error is at most $ϵ$ . Thus each eigenstate has success at least $3 / 4$ . By [F1], the total success probability is at least $(3 / 5) (3 / 4) + (2 / 5) (3 / 4) = 3 / 4$ . The eigenvalue gap is $1 / 3 - (- 1 / 2) = 5 / 6 > 1 / 2 = 2 ϵ$ , so the two intervals are disjoint.

3.1F5givenalgebra∎

The seven phase bits require the controlled powers with exponents $1, 2, 4, 8, 16, 32, 64$ . Repeating controlled- $U$ for each power uses $1 + 2 + 4 + 8 + 16 + 32 + 64 = 127$ calls.

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passOpen item page →

A bounded polynomial approximation to a scaled exponential

Example

For $H = diag (- 1 / 2, 1 / 3)$ , take $α = 1$ and target $f (x) = e^{- i x}$ on $[- 1, 1]$ . A one-ancilla exact dilation has top-left block $H$ . Decompose $f (x) = \cos x - i \sin x$ into the even component $g_{0} (x) = \cos x$ and odd component $g_{3} (x) = - \sin x$ ; the other two components are identically zero and are omitted using their exact zero certificates, so $J = {0, 3}$ . The bounded parity polynomials $p_{R} (x) = 1$ (even, degree $0$ ) and $p_{I} (x) = - x$ (odd, degree $1$ ) use exact QSP phase lists $(0)$ and $(π / 2, π / 2)$ , respectively, and give an LCU block $(I - i H) / 2$ , a $2$ -block encoding of the linear approximation to $e^{- i H}$ . In exact phase and LCU models its normalized top-block error is at most $7 / 96$ .

Facts & Assumptions

Given: $H = diag (- 1 / 2, 1 / 3)$ , $α = 1$ , exact ideal phase operations and LCU, and the functions and polynomials above.

[F1]

Finite block encoding: a unitary whose designated compression is $H$ with normalization one is an exact block encoding of $H$ .

[F2]

QSVT implementation of a bounded matrix function, Matrix-function transformation error and resource bound, QSVT action on singular subspaces, QSP signal matrix and ordered phase sequence, LCU projected-block correctness, and Uniform polynomial approximation transfers to a Hermitian matrix: parity polynomial phase lists give their stated singular transforms, the exact LCU compression is the normalized weighted sum, and scalar error on the spectrum transfers to operator norm.

[F3]

The exponential definitions of complex sine, cosine, hyperbolic sine, and hyperbolic cosine equal their entire power series and $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ : $e^{- i x} = \cos x - i \sin x$ , with the usual convergent Taylor series for sine and cosine.

[F4]

The alternating series test: if $(b_{k})$ is nonincreasing with $b_{k} \to 0$ then $\sum_{k} (- 1)^{k} b_{k}$ converges, the sum lies between any two consecutive partial sums, and the error after $n$ terms is at most $b_{n}$ : for an alternating series whose nonnegative term magnitudes decrease to zero, the remainder is bounded by the first omitted term.

Verification

1.1F1givenalgebra

Let $S = \sqrt{I - H^{2}} = diag (\sqrt{3} / 2, 2 \sqrt{2} / 3)$ and set $U = (\begin{array}{cc} H & S \\ S & - H \end{array})$ . Since $S = S^{*}$ , $H = H^{*}$ , and $H S = S H$ , $U^{*} = U$ and $U^{2} = (\begin{array}{cc} H^{2} + S^{2} & H S - S H \\ S H - H S & S^{2} + H^{2} \end{array}) = I$ . Thus $U$ is unitary and its top-left block is $H$ , proving the exact dilation assertion by [F1].

2.1F2F3step 1.1givenalgebra

The real part of $e^{- i x}$ is even and the imaginary part is $- \sin x$ , which is odd, by [F3]. For phase list $(0)$ the degree-zero signal sequence has upper-left entry $1$ . For $(π / 2, π / 2)$ , the upper-left entry of $Z (π / 2) W_{R} (x) Z (π / 2)$ is $x e^{i π} = - x$ , by the sequence definition [F2]. Using the exact dilation from step 1.1, the polynomials have matching parity and satisfy $∣ p_{R} (x) ∣ = 1$ , $∣ p_{I} (x) ∣ \leq 1$ on $[- 1, 1]$ . The coefficient weights are both one, so the LCU scale is $B = 2$ ; by [F2] its normalized weighted sum is $(I + i (- H)) / 2 = (I - i H) / 2$ , with the stated normalization.

3.1F2F3F4step 2.1algebra∎

The sine and cosine series converge by [F3], so their terms tend to zero. For $∣ x ∣ \leq 1$ , their term magnitudes are nonincreasing, so [F4] gives $∣ 1 - \cos x ∣ \leq x^{2} / 2$ ; the sine series similarly gives $∣ x - \sin x ∣ \leq ∣ x ∣^{3} / 6$ . On the two eigenvalues these imply component errors $τ_{0} \leq 1 / 8$ and $τ_{3} \leq 1 / 48$ . With $ε_{B E} = 0$ , exact phases and exact LCU, the error formula in [F2] gives normalized block error at most $(1 / 8 + 1 / 48) / 2 = 7 / 96$ . Equivalently, $∣ (1 - i x) / 2 - e^{- i x} / 2 ∣ \leq (∣ 1 - \cos x ∣ + ∣ x - \sin x ∣) / 2 \leq (x^{2} / 2 + ∣ x ∣^{3} / 6) / 2$ , which is $7 / 96$ at $x = - 1 / 2$ and $5 / 162$ at $x = 1 / 3$ . The scalar error is at most $7 / 96$ on $σ (H)$ ; applying the polynomial matrix-function transfer theorem [F2] to $p (x) = (1 - i x) / 2$ and $f (x) = e^{- i x} / 2$ gives the same operator-norm bound.

Quantum Spectral Estimation and Matrix Functions — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Spectral distribution and phase resolution for a two-level matrix

Example

Facts & Assumptions

Verification

A bounded polynomial approximation to a scaled exponential

Example

Facts & Assumptions

Verification

Sources