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An eigenphase of a unitary operator is a real number $\theta$ modulo one specifying an eigenvalue $e^{2\pi i\theta }$. Every unitary eigenvalue has modulus one. Quantum phase estimation records the eigenphase corresponding to an input eigenvector; a superposition retains the corresponding labels coherently before measurement.

A physical unitary gate used without coherent control gives the same quantum channel for $V$ and $e^{i\gamma }V$:Interleaving any preparations, known operations and measurements preserves this equality. Equivalently, each fixed measurement branch with $k$ uses gains only the overall factor $e^{ik\gamma }$, which cancels in its probability. Thus unrestricted repeated uncontrolled uses do not determine the common phase of the eigenvalues. A supplied controlled unitary gate is a different resource: $|0⟩⟨0|\otimes I+|1⟩⟨1|\otimes V$ changes by a relative control phase when $V$ is replaced by $e^{i\gamma }V$. Such controlled access permits quantum phase estimation to measure that phase relative to the identity branch.

For $\gcd (x,n)=1$ and order $r$, multiplication by $x$ modulo $n$ is a unitary operator. On the orbit of one it cyclically permutes $r$ basis states. The displayed orthonormal eigenstates have eigenvalues $e^{2\pi is/r}$, and $|1⟩=r^{-1/2}{\sum }_s|u_s⟩$. Thus quantum phase estimation on this efficiently prepared state samples a uniform $s$ without knowing $r$. Controlled powers are implemented by quantum modular exponentiation. Sufficient phase resolution allows continued-fraction recovery in quantum order finding; the reduced denominator equals $r$ only when $s$ is coprime to $r$.

For a unitary operator eigenphase $\varphi$, an $n$-qubit quantum phase estimation register of size $L=2^n$ has output probability amplitudeChoose the nearest integer label $m$ to $L\varphi /(2\pi )$, with labels understood modulo $L$, and put $d=L\varphi /(2\pi )-m$, taking the representative $|d|\le 1/2$. The finite geometric series gives $|a_m|=|\sin (\pi d)|/[L|\sin (\pi d/L)|]$. For $0<|d|\le 1/2$, concavity of sine gives $|\sin (\pi d)|\ge 2|d|$, while $|\sin (\pi d/L)|\le \pi |d|/L$. Thus the probability of the nearest label is at least $4/{\pi }^2$. At $d=0$ it is exactly one. If two labels tie for nearest, each satisfies the bound.

Let $U$ be a unitary operator and suppose initially that the target is an eigenstate $|u⟩$ with $U|u⟩=e^{2\pi i\varphi }|u⟩$, $0\le \varphi <1$. Use $t$ control qubits, so $L=2^t$. Hadamard gates prepare the uniform superposition $L^{-1/2}{\sum }_{a=0}^{L-1}|a⟩$. For each binary control position $j$, apply the controlled unitary gate $U^{2^j}$. The resulting state isApply the inverse quantum Fourier transform, with convention ${\mathrm{QFT}}_L|j⟩=L^{-1/2}{\sum }_a{e}^{2\pi iaj/L}|a⟩$, and measure the control. The exact amplitude and probability areFor an exactly representable phase $\varphi =j_0/L$, orthogonality of roots of unity makes the outcome $j_0$ certain. Otherwise the finite geometric series givesThe estimate is $j/L$, with phases interpreted modulo one.

To prove an error bound, choose a nearest label $j_0$ to $L\varphi$ on the circular grid and put $d=L\varphi -j_0$ in the interval $[-1/2,1/2]$. For $d\ne 0$, concavity of sine on $[0,\pi /2]$ gives $|\sin \pi d|\ge 2|d|$, while $L|\sin (\pi d/L)|\le \pi |d|$. Therefore the nearest-integer success bound for quantum phase estimation isHere $\Vert x{\Vert }_{\mathbb{R}/\mathbb{Z}}={\min }_{k\in \mathbb{Z}}|x-k|$. At an exact grid point the probability is one. If two nearest labels tie, either one separately has the displayed lower bound.

For high confidence one can prove a stronger quantum phase estimation tail bound. Let $d_j$ be the centered distance between $j$ and $L\varphi$, chosen with $|d_j|\le L/2$. The denominator estimate $|\sin (\pi d_j/L)|\ge 2|d_j|/L$ implies $p_j\le 1/(4d_j^2)$ away from the exact-phase case. A label at circular integer distance $\ell$ from $j_0$ has $|d_j|\ge \ell -1/2$, and there are at most two labels at each such distance. For $m\ge 2$, summing the tails givesOn the complementary event the circular phase error is at most $(m-1/2)/L$. Thus, for $b$ desired bits of accuracy, take $t=b+u$ with $u\ge 1$ and $m=2^u$ to obtainTaking $u=\lceil {\log }_2(1+1/(2\epsilon ))\rceil$ makes the right side at most $\epsilon$. This adds only $O(\log (1/\epsilon ))$ control qubits. Efficient QFT is not by itself a guarantee of efficient controlled powers of an arbitrary $U$: implementing them by repeated oracle calls uses $L-1$ calls. The modular-arithmetic application has efficient powers, as follows.

Assume $n>1$ and $\gcd (x,n)=1$, and let $r$ be the multiplicative order of $x$ modulo $n$. Define the unitary operator $U_x|y⟩=|xy\mathrm{mod}n⟩$ on $0\le y<n$, and extend it as the identity on unused computational-basis labels of the binary register. Coprimality makes multiplication a permutation, so this extension really is unitary. The orbit of $|1⟩$ consists of the $r$ distinct states $|x^j\mathrm{mod}n⟩$. Its modular multiplication eigenstates areThe last relation follows by shifting the index $j$ cyclically; orthogonality follows from the finite Fourier sum. We do not need to know $r$ to prepare one of these eigenstates: the readily prepared target obeysRunning quantum phase estimation on this state produces the mixture of the $r$ eigenphase distributions, each with weight $1/r$, because the target eigenstates are orthogonal. Thus it samples a uniformly random phase label $s$ and estimates $s/r$ with the bounds just proved. Equivalently the state before the inverse QFT is $L^{-1/2}{\sum }_a|a⟩|x^a\mathrm{mod}n⟩$, which is obtained by quantum modular exponentiation directly.

The controlled power $U_x^{2^j}$ multiplies by $x^{2^j}\mathrm{mod}n$, whose constant is computed by repeated squaring. Under the assumed efficient modular exponentiation, these controlled multiplications and the QFT use a number of gates polynomial in the register lengths. Taking $L>n^2$ requires only $O(\log n)$ phase qubits. The nearest-label event has probability at least $4/{\pi }^2$ and yields an estimate within $1/(2L)<1/(2r^2)$ of $s/r$. Since $r<n$, continued-fraction recovery in quantum order finding then identifies the reduced fraction $s/r$. The denominator is $r/\gcd (s,r)$, not necessarily $r$: when $s$ and $r$ are coprime it is the desired order, and when they are not it is only a divisor. One may test candidate denominators using modular exponentiation and repeat with fresh samples; denominators from successful samples can also be combined by a least common multiple. The probability of a nearest-label sample with $s$ coprime to $r$ is at least $(4/{\pi }^2)\phi (r)/r$. These qualifications explain exactly what the phase estimate yields without falsely assuming that the unknown order or an eigenstate was available at the start.

A central application is quantum order finding, which supplies the quantum step in the Shor algorithm for factoring an integer $M$. Choose an integer $a$ coprime to $M$, and let $r$ be its multiplicative order modulo $M$. On valid residue quantum states define the reversible modular-multiplication unitary operator $U_a|x⟩=|ax\mathrm{mod}M⟩$, extending it reversibly to any unused computational basis quantum states.

The orbit $|1⟩,|a⟩,\dots ,|a^{r-1}⟩$ has the orthonormal eigenstatesfor $s=0,\dots ,r-1$. One need not know $r$ to prepare an eigenstate: the easy input $|1⟩=r^{-1/2}{\sum }_s|u_s⟩$ makes quantum phase estimation sample one of these eigenphases. Its control register estimates $s/r$. With $N=2^t$ chosen at least of order $M^2$, the nearest quantum phase label has rational approximation error small enough for continued-fraction recovery in quantum order finding. For instance, $N>2M^2$ gives $|m/N-s/r|<1/(2r^2)$ on the nearest-label event. If $s$ and $r$ are coprime, the recovered denominator is $r$; otherwise one obtains a divisor and can repeat, combine candidates and verify them by modular exponentiation.

When the verified order is even and $a^{r/2}\not\equiv -1(\mathrm{mod}M)$, the greatest common divisorsgive nontrivial factors: their product is divisible by $M$, while neither factor inside the gcd is zero modulo $M$. Unsuccessful choices of $a$ or quantum phase samples are handled by repetition and classical verification. Controlled powers of $U_a$ are implemented by efficient reversible modular arithmetic, rather than by naively applying $U_a$ exponentially many times.

Quantum phase estimation turns modular-multiplication eigenphases into an efficiently usable period, enabling quantum factoring in time polynomial in the bit length of the integer. Factoring is important to public-key cryptography, and this is a major example of a quantum algorithm improving on known classical methods. Another use is extracting energy eigenvalues by applying quantum phase estimation to a controlled time-evolution operator, with care to avoid energy aliasing and to prepare an input having overlap with the desired eigenstate.

Label the control qubits by binary weight: the bit $x_n$ contributes $2^n$ to $x={\sum }_{n=0}^{N-1}{2}^n{x}_n$. For each $n$, use that qubit as the control of the supplied controlled-$V^{2^n}$, with the same target eigenstate $|v_j⟩$. The controlled unitary gate acts bysince $V^{2^n}|v_j⟩=e^{i{2}^n{\varphi }_j}|v_j⟩$. The target remains unchanged after every application. Multiplying these factors givesThe ordering of the tensor factors is chosen to match the binary place values. Discarding the unchanged target leaves the required phase register. This uses quantum phase kickback in the phase accumulation step of quantum phase estimation.

With access to the uncontrolled physical gate, use single-qubit process tomography. Prepare the positive eigenstate of each of $X,Y,Z$, apply $V$, and on independent copies measure each of $X,Y,Z$. Arbitrary single-qubit rotation gates let one prepare these inputs and rotate each measurement into the computational basis. The nine measured means areThey determine the real rotation matrix $R$ of the Bloch vector, and hence the action of $V$ on every qubit state. To see precisely what this identifies, suppose another unitary $U$ has the same action. Then $V^{†}U$ commutes with every Pauli matrix. Commuting with $Z$ makes it diagonal; commuting with $X$ makes its diagonal entries equal. Therefore $U=e^{i\gamma }V$.

Consequently tomography determines $V$, and its eigenvalues, up to a common phase. More explicitly, write a representative as $V=e^{i\gamma }(\cos aI-i\sin a\mathbf{n}\cdot \mathbf{\sigma })$. Its eigenvalues are $e^{i(\gamma -a)}$ and $e^{i(\gamma +a)}$, and its Bloch vector action is rotation through $2a$ about $\mathbf{n}$. The measured rotation identifies the unordered relative eigenphase information; for example $\mathrm{Tr}R=1+2\cos (2a)$. Once a representative with a specified overall phase is supplied, diagonalizing that representative gives the corresponding two eigenvalues.

The common phase itself cannot be recovered with the stated uncontrolled access. Indeed, $(e^{i\gamma }V)\rho (e^{i\gamma }V{)}^{†}=V\rho V^{†}$ for every input, including inputs entangled with ancillas. Replacing $V$ by $e^{i\gamma }V$ therefore leaves every such preparation-and-measurement experiment unchanged, whereas it multiplies both eigenvalues by $e^{i\gamma }$. For example, $I$ and $e^{i\gamma }I$ are indistinguishable physical gates under these operations. This proves that the request for absolute eigenvalue phases needs an additional phase reference or controlled-gate assumption: the global phase of an uncontrolled quantum gate is unobservable. The controlled powers in the earlier parts supply that extra resource, since $\mathrm{diag}(I,e^{i\gamma }V)$ changes a relative phase between the two control branches.

The practical advantages are that no unknown eigenstate preparation, controlled powers, entangling gates or inverse quantum Fourier transform are needed: a fixed set of single-qubit preparations and measurements suffices. The disadvantages are statistical estimation and the phase ambiguity just proved. Estimating the bounded Pauli measurement means to additive accuracy $ϵ$ at fixed confidence requires order ${ϵ}^{-2}$ independent repetitions, giving the usual sampling cost for this direct procedure. The earlier quantum phase estimation circuit uses $N$ phase qubits and high controlled powers to resolve scale $2^{-N}$ in one coherent experiment with a constant success bound; constructing those powers from controlled uses of $V$ costs $2^N-1$ gate calls, so its exponential-looking precision is not obtained with only $N$ uses of $V$. Repetitions can raise confidence. Finally, full process tomography grows exponentially with the number of system qubits, whereas the eigenstate-based phase procedure can estimate a selected eigenphase without reconstructing the entire operator.

Since $U$ is a unitary operator, its eigenstates form an orthonormal basis. Expand $|\xi ⟩={\sum }_j{c}_j|v_j⟩$, with eigenphases ${\varphi }_j=y_j/2^n$. By linearity, the unmeasured quantum phase estimation output isThis is generally an entangled state, not a phase label attached to an unchanged pure system state. The distinct eigenvalues give distinct phase labels, so a computational basis measurement yields $y_j$ with Born rule probability $|c_j{|}^2$ and leaves $|v_j⟩$ in the system register. Before measurement, all relative phases remain coherent; that is essential for the following spectral transformation. If phases were degenerate, a measured label would instead select the corresponding eigenspace component.

The usual sufficient input promises for the HHL algorithm are efficient quantum state preparation of the nonzero vector $b$, efficient access to a sparse matrix $A$ with at most $\mathrm{poly}(n)$ nonzero entries per row, and a condition number $\kappa ={\lambda }_{\max }/{\lambda }_{\min }=\mathrm{poly}(n)$ after an efficiently known normalization. Both the positions and values of the nonzero matrix elements must be computable coherently in polynomial time. More generally, efficient Hamiltonian simulation can replace the sparsity promise. We assume $A$ is invertible; if zero eigenvalues occur, one must instead restrict $b$ to their orthogonal complement and specify the desired inverse on that support.

With $a={\lambda }_{\max }$ and a known bound $\kappa$, choose $c=a/\kappa \le {\lambda }_{\min }$. The HHL controlled reciprocal rotation then succeeds with probabilityThis is the required inverse-polynomial lower bound. In the usual approximate algorithm, inverse-polynomial requested error also gives polynomial runtime under these input promises. The original HHL paper states the dependence on sparsity, conditioning, and accuracy.

There is a qualification for a literally exact version. Representability of the eigenvalues in $n$ bits does not itself make exact quantum phase estimation efficient: using $e^{2\pi iA}$ requires controlled powers corresponding to evolution times as large as $2\pi 2^{n-1}$. Under the paper's idealization we may describe their exact action, but polynomial runtime for that exact circuit additionally requires efficient implementations of those controlled powers, or equivalent efficient exact spectral access. Assuming an operation executes exactly does not bound its cost. This distinction is recorded in cost of exact phase estimation on a dyadic spectrum.

Expand $|b⟩={\sum }_i{b}_i|{\lambda }_i⟩$ in the nondegenerate eigenbasis of $A$. Apply quantum phase estimation to $U_{+}=e^{2\pi iA}$, using $U_{-}$ when inverse powers are needed, to coherently attach the eigenvalue:On an ancillary qubit perform the eigenvalue-controlled rotationUncompute the value register and measure the ancilla. Conditional on outcome one, the system is proportional toand normalization gives the desired $|\psi ⟩$.

The success probability isSince every ${\lambda }_i>0$,a positive lower bound independent of $|b⟩$. The promised precision assumptions permit the eigenvalue-controlled arithmetic and rotation.

Let the spectral decomposition of the Hermitian matrix beThe HHL algorithm proceeds as follows.

Efficiently prepare the normalized amplitude encoding $|b⟩$.
Apply quantum phase estimation to the simulated evolution $e^{iAt}$, producing an approximation of each eigenvalue: Add one ancilla and perform an eigenvalue-controlled rotationwhere $0<C\le {\min }_j|{\lambda }_j|$.
Uncompute the eigenvalue register. Conditional on measuring the ancilla as $1$, the system register isThe postselection probability can be increased with amplitude amplification.

The ingredients used here are efficient sparse Hamiltonian simulation of $e^{iAt}$ and quantum phase estimation, which converts an eigenphase of that evolution into a binary approximation of ${\lambda }_j$. For sparsity $s$, condition number $\kappa$, and error $ϵ$, the cost is polynomial in $s$, $\kappa$, $1/ϵ$, and $\log N$ under the stated access assumptions.

Finally estimate $⟨x|M|x⟩$ by repeated measurement of an efficient observable decomposition of $M$, or by a Hadamard test when $M$ is unitary. A general efficiently block-encoded Hermitian $M$ can similarly be measured through its block encoding. Repetition and a concentration inequality give additive sampling error $O(1/\sqrt{R})$ after $R$ independent preparations.

Use $|1⟩$ as the target register for quantum phase estimation of $U_a$. By part (iii), it is the equal superposition $r^{-1/2}{\sum }_s|{\psi }_s⟩$ of eigenvectors with phases $s/r$. Controlled modular multiplications implement the required powers $U_a^{2^j}$ efficiently. The phase-estimation circuit produceswhere the first register contains an $m$-bit approximation with constant success probability. A quantum measurement in the computational basis therefore outputs an approximation to $s/r$, with $s$ uniformly distributed over $\{0,\dots ,r-1\}$.

Let $L=2^t$, let $j_0$ be the nearest label to $L\varphi$, and let $m\ge 2$. The exact quantum phase estimation amplitude is $L^{-1}{\sum }_{a=0}^{L-1}{e}^{2\pi ia(\varphi -j/L)}$. The sine bound on circular distances gives probability at most $1/(4d_j^2)$, where $d_j$ is the centered distance from $j$ to $L\varphi$. At label distance $\ell$ from $j_0$, $|d_j|\ge \ell -1/2$, with at most two labels for each $\ell$. Since $(\ell -1/2{)}^{-2}\le [(\ell -1)\ell {]}^{-1}$, the tail telescopes to the displayed bound. Adding $u$ control qubits beyond $b$ desired phase bits gives circular accuracy below $2^{-b}$ with failure probability at most $[2(2^u-1){]}^{-1}$.

For a $d$-dimensional unitary operator $V={\sum }_{j=1}^d{e}^{i{\varphi }_j}|v_j⟩⟨v_j|$, a maximally mixed state $I/d$ on its target is the equal mixture of these orthonormal eigenvectors. Therefore quantum phase estimation with register size $L$ produces label $m$ with probabilityThis follows by linearity of unitary time evolution and the Born rule on the spectral mixture. If every eigenphase has an exact $L$-grid label, that label has probability equal to its multiplicity divided by $d$. Otherwise the exact distribution has tails, and the nearest labels are guaranteed only the nearest-integer success bound for quantum phase estimation weighted by the corresponding spectral probabilities.