Algorithms · Computational number theory
Pell's equation by computing the regulator
Given a positive non-square integer d, Pell's equation is x² − dy² = 1 and the goal is to find all integer solutions. The least solution can have exponentially many bits, so it cannot be written down; the computational problem is instead to compute the integer part of the regulator R = ln(x₁ + y₁√d), from which the solution can be recovered.
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Order-finding for 7 mod 15
3 counting qubits, 4 work qubits, finding the order of 7 modulo 15 (the order is 4: 7⁴ mod 15 = 1).
Step 1 of 5 · X
Probability of each outcome after this step
- 0001000100%
Prepares the work register as |0001⟩ = 1.
The whole state moves from 0000000 to 0001000 — one certain outcome before, one certain outcome after, no superposition either side.
Spreads the 3 counting qubits into an equal superposition of all 8 values.
Probability spreads from 1 outcome to 8. All 8 are equally likely.
Multiplies the work register by 7 mod 15, controlled by the first counting qubit: the k=0 controlled power of the order-finding unitary.
Probability moves: 4 outcomes appear that had none, 4 drop to zero, 8 carry probability now. Down: 0001001 12.5% → 0.0%, 0001011 12.5% → 0.0%. This is the step that entangles the qubits: from here on no single qubit has a state of its own, only the register as a whole does.
Multiplies the work register by 4 mod 15 (7 squared, mod 15), controlled by the second counting qubit: the k=1 controlled power. The k=2 power would multiply by 7⁴ mod 15 = 1, so it is the identity and is left out entirely.
Probability moves: 4 outcomes appear that had none, 4 drop to zero, 8 carry probability now. Up: 0100010 0.0% → 12.5%. Down: 0001010 12.5% → 0.0%, 0001110 12.5% → 0.0%.
An inverse QFT on the counting register reads the phase found by the controlled multiplications into a 3-bit estimate.
Probability moves: 12 outcomes appear that had none, 4 drop to zero, 16 carry probability now. Down: 0111001 12.5% → 0.0%, 0111101 12.5% → 0.0%.
Readout The counting register reads 000, 010, 100 or 110, each equally likely, corresponding to the four powers k = 0, 1, 2, 3 of the order r = 4; the work register ends entangled with the counting outcome, holding one of 1, 7, 4 or 13. Repeating this and applying the continued-fractions step of the full algorithm to a nonzero outcome recovers the order r = 4 exactly.
Open in Studio- Record typeLiterature reference
- Speedup class (secondary source)Superpolynomial
- Primary source on the speedupStated by the primary source: "The best algorithm for factoring integers has expected time L(1/3,b) for some constant b [LL93]. Assuming the GRH, the best algorithms for Pell's equation and the principal ideal problem have expected time L(1/2,b′), for some constant b′, so there is a sub-exponential gap between the best known classical algorithms."
- Reported costPolynomial in log Δ and log δ, where Δ is the quadratic discriminant and δ the precision to which the regulator is approximated, with probability exponentially close to one.
- Papers1
Given a positive non-square integer d, Pell's equation is x² − dy² = 1 and the goal is to find all integer solutions. The least solution can have exponentially many bits, so it cannot be written down; the computational problem is instead to compute the integer part of the regulator R = ln(x₁ + y₁√d), from which the solution can be recovered. Hallgren extends the hidden subgroup problem from finite groups to the group of real numbers. The regulator is an irrational period of a function defined on reduced ideals through a distance function, so the task becomes period finding where the period is not an integer and the group is not discrete. A discretized, pseudo-periodic version of that function is Fourier sampled twice over Z_q, and the two integer outputs are combined by a continued-fraction expansion to recover an integer close to the irrational period. Classical ideal composition and reduction on reduced quadratic ideals supply the function evaluations, which the paper shows can be done in time polynomial in log Δ. The framework is Shor's, applied to a group-like subset of the reals rather than to a group.
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ReturnsNothingWhat joins here
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Hallgren extends the hidden subgroup problem from finite groups to the group of real numbers. The regulator is an irrational period of a function defined on reduced ideals through a distance function, so the task becomes period finding where the period is not an integer and the group is not discrete. A discretized, pseudo-periodic version of that function is Fourier sampled twice over Z_q, and the two integer outputs are combined by a continued-fraction expansion to recover an integer close to the irrational period. Classical ideal composition and reduction on reduced quadratic ideals supply the function evaluations, which the paper shows can be done in time polynomial in log Δ. The framework is Shor's, applied to a group-like subset of the reals rather than to a group. This record's speedup class, "Superpolynomial", is a secondary source's classification of the algebraic and number theoretic algorithms it files this under — not a claim its primary paper makes. Stated by the primary source: "The best algorithm for factoring integers has expected time L(1/3,b) for some constant b [LL93]. Assuming the GRH, the best algorithms for Pell's equation and the principal ideal problem have expected time L(1/2,b′), for some constant b′, so there is a sub-exponential gap between the best known classical algorithms.". Reported cost: Polynomial in log Δ and log δ, where Δ is the quadratic discriminant and δ the precision to which the regulator is approximated, with probability exponentially close to one..
TakesNothingWhat joins here
No input port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.
Nothing in the Atlas meets this end.
ReturnsNothingWhat joins here
No output port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.
Nothing in the Atlas meets this end.
This record publishes no gate sequence and no register, so there is nothing here to read an interface off. Absent rather than empty. See all 152 →
- Record type
- Literature reference
- Speedup class (secondary source)
- Superpolynomial
- Primary source on the speedup
- Stated by the primary source: "The best algorithm for factoring integers has expected time L(1/3,b) for some constant b [LL93]. Assuming the GRH, the best algorithms for Pell's equation and the principal ideal problem have expected time L(1/2,b′), for some constant b′, so there is a sub-exponential gap between the best known classical algorithms."
- Reported cost
- Polynomial in log Δ and log δ, where Δ is the quadratic discriminant and δ the precision to which the regulator is approximated, with probability exponentially close to one.
- Speedup class
- Superpolynomial (checked against the record's own primary paper, which states it)
- Complexity basis
- section 3.2, Theorem 2 of the J. ACM version (doi:10.1145/1206035.1206039): "There is a polynomial-time quantum algorithm that, given a quadratic discriminant ∆, approximates the regulator to within δ of the associated order O in time polynomial in log∆ and logδ with probability exponentially close to one."; the supporting evaluation bound is section 2, Theorem 1, which states that the discretized function can be evaluated in time polynomial in log∆ with high probability.
- Circuit
- Not supplied
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ALGORITHM: Pell's equation by computing the regulator
PROBLEM: Given a positive non-square integer d, Pell's equation is x² − dy² = 1 and the goal is to find all integer solutions. The least solution can have exponentially many bits, so it cannot be written down; the computational problem is instead to compute the integer part of the regulator R = ln(x₁ + y₁√d), from which the solution can be recovered.
IDEA: Hallgren extends the hidden subgroup problem from finite groups to the group of real numbers. The regulator is an irrational period of a function defined on reduced ideals through a distance function, so the task becomes period finding where the period is not an integer and the group is not discrete. A discretized, pseudo-periodic version of that function is Fourier sampled twice over Z_q, and the two integer outputs are combined by a continued-fraction expansion to recover an integer close to the irrational period. Classical ideal composition and reduction on reduced quadratic ideals supply the function evaluations, which the paper shows can be done in time polynomial in log Δ. The framework is Shor's, applied to a group-like subset of the reals rather than to a group.
REPORTED COST: Polynomial in log Δ and log δ, where Δ is the quadratic discriminant and δ the precision to which the regulator is approximated, with probability exponentially close to one.
BASIS: section 3.2, Theorem 2 of the J. ACM version (doi:10.1145/1206035.1206039): "There is a polynomial-time quantum algorithm that, given a quadratic discriminant ∆, approximates the regulator to within δ of the associated order O in time polynomial in log∆ and logδ with probability exponentially close to one."; the supporting evaluation bound is section 2, Theorem 1, which states that the discretized function can be evaluated in time polynomial in log∆ with high probability.
PRIMARY SOURCE: Sean Hallgren (2007), Polynomial-Time Quantum Algorithms for Pell's Equation and the Principal Ideal Problem — https://doi.org/10.1145/1206035.1206039
This is a literature reference record, not an executable circuit.A reference record, not runnable source. Leona cannot execute it, so it cannot be saved to your Library as a circuit.
Where this sits
This record is named by the layer graph at:
- Period finding over the reals Method
Takes A circuit evaluating f on a superposition of inputs, the promise that f is periodic, the kind of object its period is (an integer in a finite cyclic group, an irrational real, a lattice of rank r), and — where the period is not an integer — the precision wanted. Returns The period: an exact integer where the group is finite, or an approximation to the requested precision together with the classical post-processing that turned the measured samples into it.
Related entries
- The principal ideal problem in a real quadratic field
Algorithms
- Shor period finding
Algorithms
- Discrete logarithm on a quantum computer
Algorithms
References
- Polynomial-Time Quantum Algorithms for Pell's Equation and the Principal Ideal Problem
Sean Hallgren · 2007
Primary source, and the source of this record's cost claim. The full text is freely available from the author's page at the Pennsylvania State University; the version read for this record is the longer journal manuscript, whose own footnote says a preliminary version appeared at STOC 2002, which is the version the Quantum Algorithm Zoo cites. Consult it for the extension of the hidden subgroup problem to the reals, for the two Fourier samplings combined by continued fractions, and for the open question in section 3.1 about whether a fixed precision suffices for all ideals.