quantrs2_core/decomposition/
solovay_kitaev.rs

1//! Solovay-Kitaev algorithm for approximating arbitrary quantum gates
2//!
3//! The Solovay-Kitaev algorithm provides an efficient method to approximate
4//! any single-qubit gate to arbitrary precision using a finite gate set
5//! (typically Clifford+T gates).
6
7use crate::error::{QuantRS2Error, QuantRS2Result};
8use crate::gate::{single::*, GateOp};
9use crate::matrix_ops::{matrices_approx_equal, DenseMatrix, QuantumMatrix};
10use crate::qubit::QubitId;
11use rustc_hash::FxHashMap;
12use scirs2_core::ndarray::{Array2, ArrayView2};
13use scirs2_core::Complex64;
14use smallvec::SmallVec;
15
16/// Configuration for the Solovay-Kitaev algorithm
17#[derive(Debug, Clone)]
18pub struct SolovayKitaevConfig {
19    /// Maximum recursion depth
20    pub max_depth: usize,
21    /// Precision goal (epsilon)
22    pub epsilon: f64,
23    /// Base gate set to use
24    pub base_set: BaseGateSet,
25    /// Cache size limit (number of sequences to store)
26    pub cache_limit: usize,
27}
28
29impl Default for SolovayKitaevConfig {
30    fn default() -> Self {
31        Self {
32            max_depth: 10,
33            epsilon: 1e-3,
34            base_set: BaseGateSet::CliffordT,
35            cache_limit: 10000,
36        }
37    }
38}
39
40/// Available base gate sets
41#[derive(Debug, Clone, Copy, PartialEq, Eq)]
42pub enum BaseGateSet {
43    /// Clifford+T gate set (H, S, T)
44    CliffordT,
45    /// V basis (H, T, X)
46    VBasis,
47    /// Custom gate set
48    Custom,
49}
50
51/// A sequence of gates
52pub type GateSequence = SmallVec<[Box<dyn GateOp>; 8]>;
53
54/// Gate sequence with its unitary matrix
55#[derive(Debug)]
56pub struct GateSequenceWithMatrix {
57    /// The gate sequence
58    pub sequence: GateSequence,
59    /// The unitary matrix of the sequence
60    pub matrix: Array2<Complex64>,
61    /// The cost (e.g., T-count for Clifford+T)
62    pub cost: usize,
63}
64
65/// Solovay-Kitaev approximator
66pub struct SolovayKitaev {
67    config: SolovayKitaevConfig,
68    /// Cache of gate sequences by recursion level
69    sequence_cache: Vec<Vec<GateSequenceWithMatrix>>,
70    /// Lookup table for finding closest sequences
71    #[allow(dead_code)]
72    lookup_table: FxHashMap<u64, Vec<usize>>,
73}
74
75impl SolovayKitaev {
76    /// Create a new Solovay-Kitaev approximator
77    pub fn new(config: SolovayKitaevConfig) -> Self {
78        let max_depth = config.max_depth;
79        let mut sequence_cache = Vec::with_capacity(max_depth + 1);
80        for _ in 0..=max_depth {
81            sequence_cache.push(Vec::new());
82        }
83
84        let mut sk = Self {
85            config,
86            sequence_cache,
87            lookup_table: FxHashMap::default(),
88        };
89        sk.initialize_base_sequences();
90        sk
91    }
92
93    /// Initialize the base gate sequences
94    fn initialize_base_sequences(&mut self) {
95        let qubit = QubitId(0);
96        let base_gates = match self.config.base_set {
97            BaseGateSet::CliffordT => {
98                vec![
99                    // Basic gates
100                    self.create_sequence_with_matrix(vec![Box::new(Hadamard { target: qubit })]),
101                    self.create_sequence_with_matrix(vec![Box::new(Phase { target: qubit })]),
102                    self.create_sequence_with_matrix(vec![Box::new(T { target: qubit })]),
103                    // Combinations
104                    self.create_sequence_with_matrix(vec![
105                        Box::new(Hadamard { target: qubit }),
106                        Box::new(Phase { target: qubit }),
107                    ]),
108                    self.create_sequence_with_matrix(vec![
109                        Box::new(Phase { target: qubit }),
110                        Box::new(Hadamard { target: qubit }),
111                    ]),
112                ]
113            }
114            BaseGateSet::VBasis => {
115                vec![
116                    self.create_sequence_with_matrix(vec![Box::new(Hadamard { target: qubit })]),
117                    self.create_sequence_with_matrix(vec![Box::new(T { target: qubit })]),
118                    self.create_sequence_with_matrix(vec![Box::new(PauliX { target: qubit })]),
119                ]
120            }
121            BaseGateSet::Custom => {
122                // User should add custom gates
123                vec![]
124            }
125        };
126
127        // Add to level 0 cache
128        for seq in base_gates {
129            if let Ok(seq) = seq {
130                self.sequence_cache[0].push(seq);
131            }
132        }
133
134        // Build sequences for higher levels
135        for level in 1..=self.config.max_depth.min(3) {
136            self.build_sequences_at_level(level);
137        }
138    }
139
140    /// Create a sequence with its matrix
141    fn create_sequence_with_matrix(
142        &self,
143        gates: Vec<Box<dyn GateOp>>,
144    ) -> QuantRS2Result<GateSequenceWithMatrix> {
145        let mut matrix = Array2::eye(2);
146        let mut cost = 0;
147
148        for gate in &gates {
149            let gate_matrix = gate.matrix()?;
150            let gate_array = Array2::from_shape_vec((2, 2), gate_matrix)
151                .map_err(|e| QuantRS2Error::InvalidInput(e.to_string()))?;
152            matrix = matrix.dot(&gate_array);
153
154            // Count T gates for cost
155            if gate.name() == "T" || gate.name() == "T†" {
156                cost += 1;
157            }
158        }
159
160        Ok(GateSequenceWithMatrix {
161            sequence: SmallVec::from_vec(gates),
162            matrix,
163            cost,
164        })
165    }
166
167    /// Build sequences at a given recursion level
168    fn build_sequences_at_level(&mut self, level: usize) {
169        if level == 0 || level > self.config.max_depth {
170            return;
171        }
172
173        let mut new_sequences = Vec::new();
174
175        // Combine sequences from lower levels
176        for i in 0..level {
177            let j = level - 1 - i;
178            let seq1_count = self.sequence_cache[i].len();
179            let seq2_count = self.sequence_cache[j].len();
180
181            for idx1 in 0..seq1_count {
182                for idx2 in 0..seq2_count {
183                    let seq1 = &self.sequence_cache[i][idx1];
184                    let seq2 = &self.sequence_cache[j][idx2];
185
186                    // Combine sequences
187                    let mut combined = SmallVec::new();
188                    combined.extend(seq1.sequence.iter().map(|g| g.clone()));
189                    combined.extend(seq2.sequence.iter().map(|g| g.clone()));
190
191                    let matrix = seq1.matrix.dot(&seq2.matrix);
192                    let cost = seq1.cost + seq2.cost;
193
194                    let new_seq = GateSequenceWithMatrix {
195                        sequence: combined,
196                        matrix,
197                        cost,
198                    };
199
200                    // Check if this is a new/better sequence
201                    if self.should_add_sequence(&new_seq, &new_sequences) {
202                        new_sequences.push(new_seq);
203                    }
204
205                    // Limit cache size
206                    if new_sequences.len() >= self.config.cache_limit / 10 {
207                        break;
208                    }
209                }
210                if new_sequences.len() >= self.config.cache_limit / 10 {
211                    break;
212                }
213            }
214        }
215
216        self.sequence_cache[level] = new_sequences;
217    }
218
219    /// Check if a sequence should be added to the cache
220    fn should_add_sequence(
221        &self,
222        new_seq: &GateSequenceWithMatrix,
223        existing: &[GateSequenceWithMatrix],
224    ) -> bool {
225        // Check if we already have a similar but better sequence
226        for seq in existing {
227            if matrices_approx_equal(&new_seq.matrix.view(), &seq.matrix.view(), 1e-10)
228                && seq.cost <= new_seq.cost
229            {
230                return false;
231            }
232        }
233        true
234    }
235
236    /// Approximate a unitary matrix using the Solovay-Kitaev algorithm
237    pub fn approximate(&mut self, target: &ArrayView2<Complex64>) -> QuantRS2Result<GateSequence> {
238        if target.shape() != &[2, 2] {
239            return Err(QuantRS2Error::InvalidInput(
240                "Target must be a 2x2 unitary matrix".to_string(),
241            ));
242        }
243
244        // Check if target is unitary
245        let target_dense = DenseMatrix::new(target.to_owned())?;
246        if !target_dense.is_unitary(1e-10)? {
247            return Err(QuantRS2Error::InvalidInput(
248                "Target matrix is not unitary".to_string(),
249            ));
250        }
251
252        // Start the recursive approximation
253        let depth = self.calculate_required_depth();
254        self.approximate_recursive(target, depth)
255    }
256
257    /// Calculate the required recursion depth based on epsilon
258    fn calculate_required_depth(&self) -> usize {
259        // The Solovay-Kitaev theorem guarantees error O(log^c(1/ε))
260        // We use a heuristic based on the desired precision
261        let log_inv_eps = (1.0 / self.config.epsilon).ln();
262        let depth = (log_inv_eps * 2.0) as usize;
263        depth.min(self.config.max_depth)
264    }
265
266    /// Recursive approximation algorithm
267    fn approximate_recursive(
268        &mut self,
269        target: &ArrayView2<Complex64>,
270        depth: usize,
271    ) -> QuantRS2Result<GateSequence> {
272        // Base case: find the closest sequence in our cache
273        if depth == 0 {
274            return self.find_closest_base_sequence(target);
275        }
276
277        // Find initial approximation at lower depth
278        let u_n_minus_1 = self.approximate_recursive(target, depth - 1)?;
279        let u_n_minus_1_matrix = self.compute_sequence_matrix(&u_n_minus_1)?;
280
281        // Compute the error
282        let error = target.to_owned() - &u_n_minus_1_matrix;
283
284        // Compute Frobenius norm manually for complex matrices
285        let mut error_norm = 0.0;
286        for val in &error {
287            error_norm += val.norm_sqr();
288        }
289        let error_norm = error_norm.sqrt();
290
291        // If error is small enough, return the approximation
292        if error_norm < self.config.epsilon {
293            return Ok(u_n_minus_1);
294        }
295
296        // Otherwise, apply group commutator method
297        self.group_commutator_correction(target, &u_n_minus_1, &u_n_minus_1_matrix, depth)
298    }
299
300    /// Find the closest base sequence to a target unitary
301    fn find_closest_base_sequence(
302        &self,
303        target: &ArrayView2<Complex64>,
304    ) -> QuantRS2Result<GateSequence> {
305        let mut best_sequence = None;
306        let mut best_distance = f64::INFINITY;
307
308        // Search through all cached sequences
309        for sequences in &self.sequence_cache {
310            for seq in sequences {
311                let diff = target.to_owned() - &seq.matrix;
312
313                // Compute Frobenius norm manually
314                let mut distance = 0.0;
315                for val in &diff {
316                    distance += val.norm_sqr();
317                }
318                let distance = distance.sqrt();
319
320                if distance < best_distance {
321                    best_distance = distance;
322                    best_sequence = Some(seq.sequence.iter().map(|g| g.clone()).collect());
323                }
324            }
325        }
326
327        best_sequence.ok_or_else(|| {
328            QuantRS2Error::ComputationError("No base sequences available".to_string())
329        })
330    }
331
332    /// Apply group commutator correction
333    fn group_commutator_correction(
334        &self,
335        target: &ArrayView2<Complex64>,
336        base_seq: &GateSequence,
337        base_matrix: &Array2<Complex64>,
338        depth: usize,
339    ) -> QuantRS2Result<GateSequence> {
340        // Compute rotation angle of the error
341        let error = target.to_owned() - base_matrix;
342        let trace = error[[0, 0]] + error[[1, 1]];
343        let angle = (trace.re / 2.0).acos();
344
345        // Find sequences V and W such that VWV†W† approximates the error rotation
346        let (v_seq, w_seq) = self.find_commutator_sequences(angle, depth - 1)?;
347
348        // Construct the corrected sequence: U_n = VWV†W†U_{n-1}
349        let mut result = SmallVec::new();
350        result.extend(v_seq.iter().map(|g| g.clone()));
351        result.extend(w_seq.iter().map(|g| g.clone()));
352        result.extend(self.compute_inverse_sequence(&v_seq)?);
353        result.extend(self.compute_inverse_sequence(&w_seq)?);
354        result.extend(base_seq.iter().map(|g| g.clone()));
355
356        Ok(result)
357    }
358
359    /// Find sequences V and W for group commutator
360    fn find_commutator_sequences(
361        &self,
362        angle: f64,
363        _depth: usize,
364    ) -> QuantRS2Result<(GateSequence, GateSequence)> {
365        // This is a simplified implementation
366        // In practice, this would involve finding specific sequences that produce the desired rotation
367        let qubit = QubitId(0);
368
369        // Use small rotations that combine to approximate the angle
370        let small_angle = angle / 4.0;
371
372        let v = vec![Box::new(RotationZ {
373            target: qubit,
374            theta: small_angle,
375        }) as Box<dyn GateOp>];
376
377        let w = vec![Box::new(RotationY {
378            target: qubit,
379            theta: small_angle,
380        }) as Box<dyn GateOp>];
381
382        Ok((SmallVec::from_vec(v), SmallVec::from_vec(w)))
383    }
384
385    /// Compute the matrix for a gate sequence
386    fn compute_sequence_matrix(
387        &self,
388        sequence: &GateSequence,
389    ) -> QuantRS2Result<Array2<Complex64>> {
390        let mut matrix = Array2::eye(2);
391
392        for gate in sequence {
393            let gate_matrix = gate.matrix()?;
394            let gate_array = Array2::from_shape_vec((2, 2), gate_matrix)
395                .map_err(|e| QuantRS2Error::InvalidInput(e.to_string()))?;
396            matrix = matrix.dot(&gate_array);
397        }
398
399        Ok(matrix)
400    }
401
402    /// Compute the inverse of a gate sequence
403    fn compute_inverse_sequence(&self, sequence: &GateSequence) -> QuantRS2Result<GateSequence> {
404        let mut inverse = SmallVec::new();
405
406        // Reverse the sequence and invert each gate
407        for gate in sequence.iter().rev() {
408            inverse.push(self.invert_gate(gate.as_ref())?);
409        }
410
411        Ok(inverse)
412    }
413
414    /// Invert a single gate
415    fn invert_gate(&self, gate: &dyn GateOp) -> QuantRS2Result<Box<dyn GateOp>> {
416        let qubit = gate.qubits()[0]; // Assuming single-qubit gates
417
418        match gate.name() {
419            "H" => Ok(Box::new(Hadamard { target: qubit })), // H is self-inverse
420            "X" => Ok(Box::new(PauliX { target: qubit })),   // X is self-inverse
421            "Y" => Ok(Box::new(PauliY { target: qubit })),   // Y is self-inverse
422            "Z" => Ok(Box::new(PauliZ { target: qubit })),   // Z is self-inverse
423            "S" => Ok(Box::new(PhaseDagger { target: qubit })),
424            "S†" => Ok(Box::new(Phase { target: qubit })),
425            "T" => Ok(Box::new(TDagger { target: qubit })),
426            "T†" => Ok(Box::new(T { target: qubit })),
427            _ => {
428                // For rotation gates, negate the angle
429                if gate.is_parameterized() {
430                    // This is a simplified approach - would need proper handling
431                    Err(QuantRS2Error::UnsupportedOperation(format!(
432                        "Cannot invert parameterized gate {}",
433                        gate.name()
434                    )))
435                } else {
436                    Err(QuantRS2Error::UnsupportedOperation(format!(
437                        "Cannot invert gate {}",
438                        gate.name()
439                    )))
440                }
441            }
442        }
443    }
444}
445
446/// Count the number of T gates in a sequence
447pub fn count_t_gates(sequence: &GateSequence) -> usize {
448    sequence
449        .iter()
450        .filter(|g| g.name() == "T" || g.name() == "T†")
451        .count()
452}
453
454/// Optimize a gate sequence by combining adjacent gates
455pub fn optimize_sequence(sequence: GateSequence) -> GateSequence {
456    let mut optimized = SmallVec::new();
457    let mut i = 0;
458
459    while i < sequence.len() {
460        if i + 1 < sequence.len() {
461            let gate1 = &sequence[i];
462            let gate2 = &sequence[i + 1];
463
464            // Check for cancellations
465            if gate1.qubits() == gate2.qubits() {
466                let combined = match (gate1.name(), gate2.name()) {
467                    ("S", "S") | ("S†", "S†") => Some("Z"),
468                    ("S", "S†") | ("S†", "S") | ("T", "T†") | ("T†", "T") | ("H", "H") => {
469                        None
470                    } // Identity
471                    _ => Some(""), // No combination
472                };
473
474                match combined {
475                    None => {
476                        // Skip both gates (they cancel)
477                        i += 2;
478                    }
479                    Some("Z") => {
480                        optimized.push(Box::new(PauliZ {
481                            target: gate1.qubits()[0],
482                        }) as Box<dyn GateOp>);
483                        i += 2;
484                    }
485                    _ => {
486                        // No optimization, keep first gate
487                        optimized.push(sequence[i].clone());
488                        i += 1;
489                    }
490                }
491            } else {
492                optimized.push(sequence[i].clone());
493                i += 1;
494            }
495        } else {
496            optimized.push(sequence[i].clone());
497            i += 1;
498        }
499    }
500
501    optimized
502}
503
504#[cfg(test)]
505mod tests {
506    use super::*;
507
508    #[test]
509    fn test_solovay_kitaev_initialization() {
510        let config = SolovayKitaevConfig::default();
511        let sk = SolovayKitaev::new(config);
512
513        // Check that base sequences are initialized
514        assert!(!sk.sequence_cache[0].is_empty());
515    }
516
517    #[test]
518    fn test_t_gate_counting() {
519        let qubit = QubitId(0);
520        let sequence: GateSequence = SmallVec::from_vec(vec![
521            Box::new(T { target: qubit }) as Box<dyn GateOp>,
522            Box::new(Hadamard { target: qubit }),
523            Box::new(TDagger { target: qubit }),
524            Box::new(Phase { target: qubit }),
525            Box::new(T { target: qubit }),
526        ]);
527
528        assert_eq!(count_t_gates(&sequence), 3);
529    }
530
531    #[test]
532    fn test_sequence_optimization() {
533        let qubit = QubitId(0);
534        let sequence: GateSequence = SmallVec::from_vec(vec![
535            Box::new(Hadamard { target: qubit }) as Box<dyn GateOp>,
536            Box::new(Hadamard { target: qubit }),
537            Box::new(Phase { target: qubit }),
538            Box::new(PhaseDagger { target: qubit }),
539        ]);
540
541        let optimized = optimize_sequence(sequence);
542        assert_eq!(optimized.len(), 0); // All gates should cancel
543    }
544}
quantrs2_core/decomposition/solovay_kitaev.rs

quantrs2_core/decomposition/
solovay_kitaev.rs
