JOURNAL OF MODERN OPTICS, 2018 https://doi.org/10.1080/09500340.2017.1423123

Efficient eigenvalue determination for arbitrary Pauli products based on generalized spin-spin interactions

D. Leibfried and D. J. Wineland

Ion Storage Group, Time and Frequency Division, National Institute of Standards and Technology, Boulder, CO, USA

ABSTRACT ARTICLE HISTORY Effective spin-spin interactions between N + 1 qubits enable the determination of the eigenvalue Received 30 June 2017 of an arbitrary Pauli product of dimension N with a constant, small number of multi-qubit gates Accepted 20 December that is independent of N and encodes the eigenvalue in the measurement basis states of an extra 2017 ancilla qubit. Such interactions are available whenever qubits can be coupled to a shared harmonic KEYWORD oscillator, a situation that can be realized in many physical qubit implementations. For example, Quantum information suitable interactions have already been realized for up to 14 qubits in ion traps. It should be possible to implement stabilizer codes for quantum error correction with a constant number of multi-qubit gates, in contrast to typical constructions with a number of two-qubit gates that increases as a function of N. The special case of finding the parity of N qubits only requires a small number of operations that is independent of N. This compares favorably to algorithms for computing the parity on conventional machines, which implies a genuine quantum advantage.

1. Preface plemented if the qubits are coupled to a shared harmonic oscillator, a condition realized in several physical qubit Some of the work presented here was developed in 2004 implementations. For trapped ion qubits, suitable inter- to support a spectroscopy demonstration with three en- actions that couple all N + 1 spins simultaneously in one tangled ions (1), where a special case of the key relation operation were first discussed for robust entanglement Equation (6) below was stated without proof and with- operations typically referred to as Mømer-Sørensen-type out discussing further implications. This publication ap- gates (2–5). Such operations form the basis for most of peared before supplemental materials became common the recent work with entangled states of ions in quantum and the proof was too long to include it in a letter-style information processing and quantum simulation. For publication at the time. Thirteen years later, we have example, work described in (1, 6–13) produced entangled picked up the thread again and generalize the special case states by using lasers to couple the qubits to a harmonic stated in (1) to relate more general spin-spin interactions normal-mode motion of the ions and in (14–17)this to arbitrary Pauli product operators. was accomplished with microwave fields. These and re- It may be rare that experimental physicists resort to lated experiments encompass a wide range of contexts, the principle of mathematical induction, as we have done including precision spectroscopy, quantum information below. However, our contribution seems to be in tune processing and quantum simulation. In particular, the with a particularly important lesson that Danny’s life can effective spin-spin operations at the heart of our study teach us: You should not be afraid to do things in your have been realized for up to 14 ion-qubits (13)andhave own way, no matter how far of the beaten path this might enabled the highest fidelity two-qubit gates reported to take you. date at F 0.999 (18, 19). The natural extensibility of Mølmer-Sørensen type operations to more than two 2. Introduction qubits can offer algorithmic advantages over the more conventional decomposition into two-qubit gates and In this contribution we provide a link between effective single qubit rotations. spin-spin interactions of N +1 qubits and the eigenvalues Here we show, that one or two operations of this of an arbitrary Pauli product of dimension N.Suitable type, on N qubits and one ancilla-qubit, together with effective spin-spin interactions can, for example, be im-

CONTACT D. Leibfried [email protected] We dedicate this work to Danny Segal, scientist, community builder and friend. Your bold and cheerful way of leaving the trodden paths and pursuing yourown goals have made our field richer. We will not be able to fill your void, but we will carry on. © 2018 Informa UK Limited, trading as Taylor & Francis Group 2 D. LEIBFRIED AND D. J. WINELAND

2 individual single-qubit rotations that can be performed Uk = exp[−iχJk ],(3) in parallel, are sufficient to determine the eigenvalue of an arbitrary Pauli-product operator on the group of N of the that acts uniformly on all qubits with a suitable coupling qubits. The eigenvalue is stored it in the measurement ba- parameter χ can be implemented in trapped ion systems sis eigenstates of the ancilla-qubit after these operations. (see also (5)). The approach in ion traps can be general- Read-out of the ancilla state projects the N qubits into a ized to any system of qubits that couples uniformly to one corresponding eigenstate of the Pauli-product operator harmonic oscillator. The theoretical work set off a flurry that is available for further processing. Therefore, such of experiments in which entangled states of two to 14 ion operations may become useful in error correction, as qubits were produced and studied. Examples of this effort discussed in more detail below. were briefly mentioned in the introduction (1, 6–17). The parity of a state of N ions can be determined Here we generalize Uk, in which interactions are uni- with a total of three (N even) or four operations (N odd) formly along direction k over all qubits, to allow for and is revealed by one measurement on the ancilla-qubit, interactions along different directions on different qubits. independent of the size of N. This constitutes a constant In particular, for every product of Pauli matrices and the depth parity-finding algorithm with a circuit of a size identity over N qubits (kl ={i, x, y, z}), proportional to N, while parity determination algorithms on conventional computers require polynomial sized cir- N P = σ cuit families of close to log (N) depth (20). Our proposed N l,kl ,(4) implementation therefore provides an advantage even for l=1 relatively small N and falls into a family of constant depth we define an associated operator DN parity circuits discussed in (21). N D = / σ N 1 2 l,kl ,(5) 3. Generalized spin-spin interactions and Pauli l=1 products that can be used as the generator of a generalized spin- We consider N two-level systems with logical basis of spin D2 -operation according to GN = exp[−iαD2 ]. the l-th qubit (l {1, ..., N}) defined by the eigenstates N N The connection between PN and DN will be discussed in {| ↓, |↑}of the z-component of a spin-1/2 angular Section 4. Products of Pauli-matrices like PN are essential momentum operator Sl k with k ={x, y, z}. We write σl i , , as so called stabilizers in a certain class of quantum-error for the 2 × 2 identity matrix in the state space of the l- correction codes (22) (see section 5). In the special cases th qubit and σl,k for the Pauli-matrices for k ={x, y, z}. kl = k for all l, the operator DN is equivalent to Jk, When setting  = 1 for simplicity, the eigenvalues in the therefore all relations that hold for DN will also be true measurement basis are for any Jk of dimension N. 1 1 Sl z|↑= σl z|↑= |↑, , 2 , 2 4. Connection between generalized spin-spin S |↓=1σ |↓=−1|↓ l,z l,z ,(1)interactions and Pauli product operators 2 2 With the above definitions, we now state our main result: Our previous work relied on the collective angular mo- Depending on whether N is odd or even, the correspond- mentum operator in the N-spin Bloch vector ing DN and PN fulfill representation  [−i π D2 ];N N N exp N even   UN = π2 π   exp[−i DN ] exp[−i D2 ];N odd J = Sl = 1/2 σl. (2) 2 2 N ( − i π )   l=1 l=1 exp E N+E = √ 4 1 + i PN (6) 2 This notation allows, for example, for a compact rep- (N) resentation of the collective rotation operator Rk ≡ with E = 1forN even and E = 2forN odd. Equation [−i π J ] k ={x y z} π/ exp 2 k ( , , ), often called a ‘ 2’-pulse, (6) was stated without proof for the uniform cases where applied to all N qubits uniformly. kl = k as Equation (2) in (1).Inthespecialcaseof Mølmer and Sørensen (2, 3) (for k = φ and σl,φ = kl = k = x and applying UN to the initial state |↓ cos (φ)σl,x − sin (φ)σl,y) and Milburn (4) (for k = z) , N≡|↓, ↓, ..., ↓,relation(6) was also given by showed how the interaction Sørensen and Mølmer (2). The fact that Equation (6) JOURNAL OF MODERN OPTICS 3

(iπ/ )   applies to arbitrary input states is crucial for the efficient exp 4 N+3 = √ 1 + i σ1 ...σN+2 . (10) algorithms we present in Section 5, including a constant 2 depth algorithm for finding the parity of a state. It should N + N also be mentioned that since Ref. (1) appeared, another Therefore Equation (6) is true for 2( even) if related constant depth algorithm for finding the parity we can show that 23 was published ( ). This work does not contain a proof N    J2 1 of Equation (6) and relies on two -operations instead 1 − iσka 1 − iσkb 2 of one. k=1 l σ   For all , all operators l,kl commute with each other 1 N N σ 2 = = 1 + i σ1 ...σN − ab + i σ1 ...σN ab , (11) and l,kl 1 holds for each of them. To simplify the 2 σ ≡ σ notation during the proof, we abbreviate the l,kl l. As a first step in the proof we can rewrite the exponential when a2 = b2 = 1. This can be shown by another α GN (−i σ σ ) = (α/ )− N = in as a product and use exp 2 k l cos 2 inductive proof. Again for 2, relation (11)iseasily i sin (α/2)σkσl to arrive at verified. For N replaced by N + 2weget

α N N+2    2 1 exp ( − iαDN ) = exp ( − i N) 1 − iσka 1 − iσkb 4 l= 2 2 k=1   l−1 N    1    × cos (α/2) − i sin (α/2)σkσl . = 1 − iσka 1 − iσkb 2 k=1 k=1 (7) 1   × 1 − iσN+1(a + b) − ab π 4  For α = we obtain 2 × 1 − iσN+2(a + b) − ab   N l− 1 N N π π  1   = 1 + i σ1 ...σN − ab + i σ1 ...σN ab 2 1 exp ( − i D ) = exp ( − i N) √ 1 − iσkσl . 4   N × − σ σ − ab − σ σ ab 2 8 l= k= 2 1 N+1 N+2 N+1 N+2 2 1   (8) 1 N+ = 1 − ab + (1 + ab)i 2σ ...σN , (12) Starting from this result we can now prove Equation (6) 2 1 for N even or odd separately by induction. which completes the argument for N even. 4.1. Even N 4.2. Odd N Equations (6)and(8) are obviously equivalent for N = 2.

To show equivalence of equations (6)and(8)forN = For N odd and N ≥ 3 the relation for UN corresponding N +2 we re-express UN+2 using Equation (8) as a product to Equation (8)is of UN times additional factors, and substitute the right π π UN 2 hand side of Equation (8)for . UN = exp ( − i DN ) exp ( − i DN ) 2 2 N l− π N+2 l−1   π  1   1 = ( − i N) √1 − iσ σ UN+2 = exp ( − i (N + 2)) √ 1 − iσkσl exp 1 k l 8 8 2 l=2 k=1 2 l=2 k=1 ( − iπ/ ) N exp 2 N+1 1   = (1 + i σ1 ...σN × √ − iσ . 2 1 m (13) N m=1 2 − iσN+1σN+2 + i σ1 ...σN+2) N  For N = 3 the equivalence of Equation (6) and Equa- × 1( − iσ σ )( − iσ σ ). 1 k N+1 1 k N+2 (9) tion (13) can be shown by explicit calculation. For N = k= 2 1 N + 2 the proof proceeds similarly to the even case. One On the other hand we can verify that can first show that π π ( − i D ) ( − i D2 ) 1 N+1 N exp N+2 exp N+2 (1 + i σ1 ...σN − iσN+1σN+2 + i σ1 ...σN+2) 2 2 2 −i π e 8   1 N N = √ − iN σ ...σ × (1 + i σ ...σN − σN+ σN+ + i σ ...σN+ ) 1 1 N 2 1 1 2 1 2 2 4 D. LEIBFRIED AND D. J. WINELAND

  × 1 − σ − σ − σ σ the algorithm we add one ancilla qubit that we prepare in 1 N+1 N+2 N+1 N+2 N+ 2 |↑. In the following we assume that N + 1and 1 are N 2  1 even. Implementation of cases with N +1 odd will require × ( − iσ σ )( − iσ σ ). N+ 1 k N+1 1 k N+2 (14) additional single-qubit rotations and if 1 is not even, k= 2 2 1 this will produce a different sign between the identity operator and the parity operator in Equation (17), which Analogously to the even case, one can then prove by can be taken into account by changing the axis of the final a2 = b2 = induction that for 1 rotation on the ancilla in Equation (19) below. For our N particular choice of N + 1, UN+1 simplifies to  1   (1 − a − b − ab) 1 − iσk(a + b) − ab π e−i   k= 2 4 N+  1  UN+ = √ + i 2σ ...σN+ N 1 1 1 1 = 1 − ab + (1 + ab)i σ1 ...σN . (15) 2 −i π e 4   = √ 1 + iPN σN+1 . (17) Since 2 −i π   e 8 N We assume the ancilla at position N + 1 was prepared in √ 1 − i σ1 ...σN (1) 2 state |↑and first apply a π/2-rotation Ry to the ancilla   1 N N only and then UN+1 as it appears on the right-hand side × 1 + i σ ...σN+ + i σ ...σN − σN+ σN+ 2 1 2 1 1 2 of Equation (17)toallN +1 qubits, including the ancilla, −i π e 8   for which we choose σN+ = σz.Thisresultsin = √ − iN+2σ ...σ 1 1 1 N+2 , (16) 2 (1) UN+1 |p(Ry |↑) the proof is complete. exp[−iπ/4]   1 = √ 1 + iPN σz |p√ (|↑+|↓) 2 2  [−iπ/ ] + ip − ip 5. Efficient stabilizer-code syndrome =|pexp √ 4 1√ |↑+1√ |↓ . measurement 2 2 2 (18) Stabilizer codes are a powerful tool for quantum error- correction and extensively discussed in the literature. (R(1))† An introduction to this subject and many of the early A final rotation x of the ancilla qubit around the x original references can be found in (22). Here we only -direction will produce the state need a few basic facts, namely that errors are detected by (1) † (1) p{− } (Rx ) UN+1 |p(Ry |↓) determining the eigenvalues 1, 1 of Pauli products  that are designed to detect the absence or presence of a 1 + p 1 − p P =|p |↑+ |↓ . (19) certain error. The Pauli product N is then called a sta- 2 2 bilizer and can be applied to a state. For a state |p inside the code space, the eigenvalue, also called the syndrome Thesyndromeof|p, is now encoded in the state of is PN |p=+1|p, while measuring PN |p=−1|p the ancilla which will be |↑for p = 1or|↓for p =−1 indicates that the code state has been compromised by and can be read out without disturbing the code state |p. a specific error that the stabilizer code is constructed If the input state to this pulse sequence is not an to detect. By taking advantage of interference, we can eigenstate of the stabilizer PN , finding the ancilla in the determine the syndrome of the stabilizer PN acting on state corresponding to p will project the initial state into a string of N qubits based on the generalized spin-spin a superposition of eigenstates |Sp with eigenvalue p. interactions mediated by the associated operators DN . If p = 1, |S1 contains at least one code state of the Implementations require a constant number of multi- stabilizer code. Otherwise, one can take note that p =−1, qubit operations and possibly a number of single qubit which means that |S−1 contains at least one state of the rotations that is linear in N, depending on details of code rotated by one of the errors that yield p =−1 the implementation and the stabilizers in the code (see when PN is applied. After applying all error operators Section 6). The read-out requires only one ancilla and and learning their syndromes, the resulting state can be does not disturb the code state |p which can therefore recovered into the codes space by applying the correction remain encoded for further steps. operation compatible with all the syndrome results one We assume that N qubits are in an eigenstate |p of a has found. To exploit this for code-state preparation, we certain stabilizer PN , PN |p=p|p,withp{−1, 1}.For can initialize the N qubits in an equal superposition of JOURNAL OF MODERN OPTICS 5

(N) m † † † π/ R (D ) = R Jz(R R )Jz(R R )...JzR all states by applying a collective 2-rotation y and N k,N k,N k,N k,N k,N k,N then successively measure all stabilizers PN that are error † m = Rk N (Jz) Rk,N . (20) operators of the code. We keep track of all instances of , p =− measuring 1 and eventually apply the correction Because this reasoning can be applied to every term in operation corresponding to the set of syndromes we have the sum defining the operator-exponential function, we |S(f ) found. After this correction, the resulting state 1 must can initially apply Rk,N ,thenUz and then undo the initial † † be a superposition of code states and can be projected R G = R UzR rotation with k,N , which implies N k,N k,N . into a certain encoded qubit state by, for example, finally After this we can recombine all N qubits and have ef- encoded σ¯ measuring the eigenvalue of the operator z fectively implemented GN up to a global phase, because P which is implemented by another stabilizer σ¯z .Inthis GN = exp[iα(N − N )]GN . Any other uniform J2 inter- way, an encoded qubit state can be efficiently produced in action Uφ can also be used for the uniform coupling of all N steps. In principle this scheme can be executed several N qubits, after modifying the initial and final rotations times to increase the probability of obtaining a code state, to match the direction specified by φ. even if not all operations and measurements are perfect. Separation of qubits is not required in an architecture In the special case of measuring the parity operator PN , asdescribedin(26), where, in a minimal construction, 2 2 by using the uniform Mølmer-Sørensen type interaction Mølmer and Sørensen Jx or Jy type interactions that act DN = Jz, the parity can be determined with a constant globally on all N qubits can be supplemented by individ- N N circuit depth of three ( even) or four operations ( odd) ually addressed σl,z rotations, applied to only the qubits and one measurement on the ancilla qubit, a number of that are supposed to not partake in the J2-interaction. The operations that only depend on N being odd or even, individual rotations are realized by AC-Stark shifts and but not on the size of N. Parity finding algorithms on refocus each of the addressed qubits in such a way that conventional computers require polynomial sized circuit their state is unchanged by the total operation. Depend- families of close to log N depth (20). The parity finding ing on the particular structure of the error-correction algorithm proposed here therefore provides an advantage code, a more optimal sequence in the sense defined in even for relatively small N and is a member of family of (26) may also exist, but will be hard to find using the constant depth parity circuits discussed in (21). numerical optimization methods described in this work as the state space of the code increases in size. In any 6. Implementations in ion trap systems case, this type of implementation should be particularly efficient if all stabilizers contain the same Pauli-matrix In many codes, for example CSS codes (22), stabilizers apart from the individually refocused group of qubits, as σ σ are products of only one Pauli-operator, either l,x or l,y, in CSS codes. σ and the identity l,i. For such codes, stabilizer measure- Alternatively, one can use an array of tightly focused ments with just Mølmer and Sørensen type interactions laser beams that address ions individually (27) and allow φ = σl φ = σl x φ =−π/ σ with either 0for , , or 2for for precise definition of individual operations l,kl on σφ = σ l,y seems possible. The third Pauli operator is not each ion (with σl,i corresponding to not turning the indi- σ required because a l,z (phase-flip) error can be thought vidual beam on for the l-th ion). With such a setup, any σ σ 2 of as sequentially occurring l,x and l,y (bit-flip) errors. DN interaction can be implemented directly and executed The conceptually simplest way to realize identities in a constant number of parallel operations (constant could be to physically remove all qubits that are acted on depth). by σl,i within DN (sonophysicaloperationonthatqubitis required), for example by transporting them to a location Acknowledgements away from where interactions are applied in a multi- zone architecture (24, 25). This effectively reduces the This work was mostly developed in the environment of the original stabilizer to PN with N ≤ N and all elements in NIST Ion Storage Group, which therefore owns a great deal of the credit. We would like to thank Jim Bergquist, John the stabilizer PN are Pauli-matrices. We can then either Bollinger, James Chou, David Hume, Wayne Itano, David apply Mølmer and Sørensen type interactions with two Leibrandt and Andrew Wilson as well as all the post-docs, different phases, or, for non-CSS codes that may require grad students and summer students and the administrative and more complicated Pauli products, rotate each remaining technical staff of the Time and Frequency Division at NIST. In r σ = r†σ r qubit individually by l such that l,kl l l,z l.Ifwe addition, we gratefully acknowledge numerous discussions and denote the operator that applies such individual rotations useful advice from Manny Knill and the members of his group. N This paper is a contribution of NIST and is not subject to US R = r D = to all remaining qubits as k,N l=1 l we have N † copyright. R JzR m D k,N k,N and can rewrite any integer power of N as 6 D. LEIBFRIED AND D. J. WINELAND

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